Theorem elsetchom

Description: A morphism of sets is a function. (Contributed by Mario Carneiro, 3-Jan-2017)

	Ref	Expression
Hypotheses	setcbas.c	$$⊢C=\mathrm{SetCat}\left(U\right)$$
	setcbas.u	$$⊢\phi \to U\in V$$
	setchomfval.h	$$⊢H=\mathrm{Hom}\left(C\right)$$
	setchom.x	$$⊢\phi \to X\in U$$
	setchom.y	$$⊢\phi \to Y\in U$$
Assertion	elsetchom	$$⊢\phi \to \left(F\in \left(XHY\right)\leftrightarrow F:X⟶Y\right)$$

Proof

Step	Hyp	Ref	Expression
1		setcbas.c	$$⊢C=\mathrm{SetCat}\left(U\right)$$
2		setcbas.u	$$⊢\phi \to U\in V$$
3		setchomfval.h	$$⊢H=\mathrm{Hom}\left(C\right)$$
4		setchom.x	$$⊢\phi \to X\in U$$
5		setchom.y	$$⊢\phi \to Y\in U$$
6	1 2 3 4 5	setchom	$$⊢\phi \to XHY=Y^X$$
7	6	eleq2d	$$⊢\phi \to \left(F\in \left(XHY\right)\leftrightarrow F\in \left(Y^X\right)\right)$$
8	5 4	elmapd	$$⊢\phi \to \left(F\in \left(Y^X\right)\leftrightarrow F:X⟶Y\right)$$
9	7 8	bitrd	$$⊢\phi \to \left(F\in \left(XHY\right)\leftrightarrow F:X⟶Y\right)$$