MHB What Determines When Two Mathematical Functions Are Equal?

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The discussion centers on the proof that two functions, f and g, are equal if and only if they have the same domain and produce the same output for every input in that domain. The proof demonstrates that if f equals g, then their domains must also be equal, and vice versa. It establishes that if the domains are equal and the functions yield the same values for all inputs, then each function is a subset of the other. The user seeks clarification specifically on the part of the proof that shows the reverse implication. Understanding this aspect is crucial for grasping the overall proof of function equality.
evinda
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Hello! (Wave)

I am looking at the proof of the following sentence:

Sentence:

Let $f,g$ functions. Then:

$$f=g \leftrightarrow dom(f)=dom(g) \wedge (\forall x \in dom(f)) (f(x)=g(x))$$

Proof:

$$\Rightarrow$$
If $f=g$, then $f \subset g$ and $g \subset f \Rightarrow dom(f) \subset dom(g) \text{ and } dom(g) \subset dom(f)$

So, $dom(f)=dom(g)$.

Also, from the sentence: If $f,g$ functions and $f \subset g$, we have that $\forall x \in dom(f): f(x)=g(x)$
we have that: $\forall x \in dom(f) f(x)=g(x)$

$$\Leftarrow$$

We want to show $f=g$, knowing that $dom(f)=dom(g)$ and $\forall x \in dom(f) f(x)=g(x)$

It suffices to show that $f \subset g$ and $g \subset f$.

Let $t \in f$. Then, $t=<x,f(x)>$, for a $x \in dom(f)$.
But, $dom(f)=dom(g)$ and so:
$$x \in dom(g) \leftrightarrow <x,g(x)> \in g$$
From the hypothesis, $f(x)=g(x)$ and so $t=<x,f(x)> \in g$.

Therefore, $f \subset g$.

In the same way, we show that $g \subset f$.
Therefore, $f=g$.

Could you explain me the $\Leftarrow$ part of the proof? (Thinking)
 
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