funcfn2 - Metamath Proof Explorer

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Theorem funcfn2 17887

Description: The morphism part of a functor is a function. (Contributed by Mario Carneiro, 3-Jan-2017.)

**Hypotheses**
Ref	Expression
funcfn2.b	⊢ 𝐵 = (Base‘𝐷)
funcfn2.f	⊢ (𝜑 → 𝐹(𝐷 Func 𝐸)𝐺)

**Assertion**
Ref	Expression
funcfn2	⊢ (𝜑 → 𝐺 Fn (𝐵 × 𝐵))

Proof of Theorem funcfn2 Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		funcfn2.b	. . 3 ⊢ 𝐵 = (Base‘𝐷)
2		eqid 2736	. . 3 ⊢ (Hom ‘𝐷) = (Hom ‘𝐷)
3		eqid 2736	. . 3 ⊢ (Hom ‘𝐸) = (Hom ‘𝐸)
4		funcfn2.f	. . 3 ⊢ (𝜑 → 𝐹(𝐷 Func 𝐸)𝐺)
5	1, 2, 3, 4	funcixp 17885	. 2 ⊢ (𝜑 → 𝐺 ∈ X𝑥 ∈ (𝐵 × 𝐵)(((𝐹‘(1^st ‘𝑥))(Hom ‘𝐸)(𝐹‘(2^nd ‘𝑥))) ↑_m ((Hom ‘𝐷)‘𝑥)))
6		ixpfn 8922	. 2 ⊢ (𝐺 ∈ X𝑥 ∈ (𝐵 × 𝐵)(((𝐹‘(1^st ‘𝑥))(Hom ‘𝐸)(𝐹‘(2^nd ‘𝑥))) ↑_m ((Hom ‘𝐷)‘𝑥)) → 𝐺 Fn (𝐵 × 𝐵))
7	5, 6	syl 17	1 ⊢ (𝜑 → 𝐺 Fn (𝐵 × 𝐵))

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1540 ∈ wcel 2109 class class class wbr 5124 × cxp 5657 Fn wfn 6531 ‘cfv 6536 (class class class)co 7410 1^st c1st 7991 2^nd c2nd 7992 ↑_m cmap 8845 Xcixp 8916 Basecbs 17233 Hom chom 17287 Func cfunc 17872

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2708 ax-rep 5254 ax-sep 5271 ax-nul 5281 ax-pow 5340 ax-pr 5407 ax-un 7734

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2810 df-nfc 2886 df-ne 2934 df-ral 3053 df-rex 3062 df-rab 3421 df-v 3466 df-sbc 3771 df-csb 3880 df-dif 3934 df-un 3936 df-in 3938 df-ss 3948 df-nul 4314 df-if 4506 df-pw 4582 df-sn 4607 df-pr 4609 df-op 4613 df-uni 4889 df-iun 4974 df-br 5125 df-opab 5187 df-mpt 5207 df-id 5553 df-xp 5665 df-rel 5666 df-cnv 5667 df-co 5668 df-dm 5669 df-rn 5670 df-iota 6489 df-fun 6538 df-fn 6539 df-f 6540 df-fv 6544 df-ov 7413 df-oprab 7414 df-mpo 7415 df-map 8847 df-ixp 8917 df-func 17876

This theorem is referenced by: funcoppc 17893 cofuval 17900 cofulid 17908 cofurid 17909 prf1st 18221 prf2nd 18222 1st2ndprf 18223 curfuncf 18255 uncfcurf 18256 curf2ndf 18264 oppfvallem 49048 uptposlem 49097 diag1 49182 termcfuncval 49384

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