Theorem fulli

Description: The morphism map of a full functor is a surjection. (Contributed by Mario Carneiro, 27-Jan-2017)

	Ref	Expression
Hypotheses	isfull.b	$$⊢B={\mathrm{Base}}_C$$
	isfull.j	$$⊢J=\mathrm{Hom}\left(D\right)$$
	isfull.h	$$⊢H=\mathrm{Hom}\left(C\right)$$
	fullfo.f	$$⊢\phi \to F\left(C\mathrm{Full}D\right)G$$
	fullfo.x	$$⊢\phi \to X\in B$$
	fullfo.y	$$⊢\phi \to Y\in B$$
	fulli.r	$$⊢\phi \to R\in \left(F\left(X\right)JF\left(Y\right)\right)$$
Assertion	fulli	$$⊢\phi \to \exists f\in \left(XHY\right)R=\left(XGY\right)\left(f\right)$$

Proof

Step	Hyp	Ref	Expression
1		isfull.b	$$⊢B={\mathrm{Base}}_C$$
2		isfull.j	$$⊢J=\mathrm{Hom}\left(D\right)$$
3		isfull.h	$$⊢H=\mathrm{Hom}\left(C\right)$$
4		fullfo.f	$$⊢\phi \to F\left(C\mathrm{Full}D\right)G$$
5		fullfo.x	$$⊢\phi \to X\in B$$
6		fullfo.y	$$⊢\phi \to Y\in B$$
7		fulli.r	$$⊢\phi \to R\in \left(F\left(X\right)JF\left(Y\right)\right)$$
8	1 2 3 4 5 6	fullfo	$$⊢\phi \to \left(XGY\right):XHY\underset{onto}{⟶}F\left(X\right)JF\left(Y\right)$$
9		foelrn	$$⊢\left(\left(XGY\right):XHY\underset{onto}{⟶}F\left(X\right)JF\left(Y\right)\wedge R\in \left(F\left(X\right)JF\left(Y\right)\right)\right)\to \exists f\in \left(XHY\right)R=\left(XGY\right)\left(f\right)$$
10	8 7 9	syl2anc	$$⊢\phi \to \exists f\in \left(XHY\right)R=\left(XGY\right)\left(f\right)$$