Theorem funcfn2 17920

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 2735	. . 3 ⊢ (Hom ‘𝐷) = (Hom ‘𝐷)
3		eqid 2735	. . 3 ⊢ (Hom ‘𝐸) = (Hom ‘𝐸)
4		funcfn2.f	. . 3 ⊢ (𝜑 → 𝐹(𝐷 Func 𝐸)𝐺)
5	1, 2, 3, 4	funcixp 17918	. 2 ⊢ (𝜑 → 𝐺 ∈ X𝑥 ∈ (𝐵 × 𝐵)(((𝐹‘(1st ‘𝑥))(Hom ‘𝐸)(𝐹‘(2nd ‘𝑥))) ↑m ((Hom ‘𝐷)‘𝑥)))
6		ixpfn 8942	. 2 ⊢ (𝐺 ∈ X𝑥 ∈ (𝐵 × 𝐵)(((𝐹‘(1st ‘𝑥))(Hom ‘𝐸)(𝐹‘(2nd ‘𝑥))) ↑m ((Hom ‘𝐷)‘𝑥)) → 𝐺 Fn (𝐵 × 𝐵))
7	5, 6	syl 17	1 ⊢ (𝜑 → 𝐺 Fn (𝐵 × 𝐵))

Syntax hints:   → wi 4   = wceq 1537   ∈ wcel 2106   class class class wbr 5148   × cxp 5687   Fn wfn 6558  ‘cfv 6563  (class class class)co 7431  1^st c1st 8011  2^nd c2nd 8012   ↑_m cmap 8865  Xcixp 8936  Basecbs 17245  Hom chom 17309   Func cfunc 17905

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-map 8867  df-ixp 8937  df-func 17909

This theorem is referenced by:  funcoppc  17926  cofuval  17933  cofulid  17941  cofurid  17942  prf1st  18260  prf2nd  18261  1st2ndprf  18262  curfuncf  18295  uncfcurf  18296  curf2ndf  18304