Theorem funcfn2 17780

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 2733	. . 3 ⊢ (Hom ‘𝐷) = (Hom ‘𝐷)
3		eqid 2733	. . 3 ⊢ (Hom ‘𝐸) = (Hom ‘𝐸)
4		funcfn2.f	. . 3 ⊢ (𝜑 → 𝐹(𝐷 Func 𝐸)𝐺)
5	1, 2, 3, 4	funcixp 17778	. 2 ⊢ (𝜑 → 𝐺 ∈ X𝑥 ∈ (𝐵 × 𝐵)(((𝐹‘(1st ‘𝑥))(Hom ‘𝐸)(𝐹‘(2nd ‘𝑥))) ↑m ((Hom ‘𝐷)‘𝑥)))
6		ixpfn 8835	. 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 1541   ∈ wcel 2113   class class class wbr 5095   × cxp 5619   Fn wfn 6483  ‘cfv 6488  (class class class)co 7354  1^st c1st 7927  2^nd c2nd 7928   ↑_m cmap 8758  Xcixp 8829  Basecbs 17124  Hom chom 17176   Func cfunc 17765

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2705  ax-rep 5221  ax-sep 5238  ax-nul 5248  ax-pow 5307  ax-pr 5374  ax-un 7676

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2725  df-clel 2808  df-nfc 2882  df-ne 2930  df-ral 3049  df-rex 3058  df-rab 3397  df-v 3439  df-sbc 3738  df-csb 3847  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4283  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4861  df-iun 4945  df-br 5096  df-opab 5158  df-mpt 5177  df-id 5516  df-xp 5627  df-rel 5628  df-cnv 5629  df-co 5630  df-dm 5631  df-rn 5632  df-iota 6444  df-fun 6490  df-fn 6491  df-f 6492  df-fv 6496  df-ov 7357  df-oprab 7358  df-mpo 7359  df-map 8760  df-ixp 8830  df-func 17769

This theorem is referenced by:  funcoppc  17786  cofuval  17793  cofulid  17801  cofurid  17802  prf1st  18114  prf2nd  18115  1st2ndprf  18116  curfuncf  18148  uncfcurf  18149  curf2ndf  18157  oppfvallem  49263  uptposlem  49325  diag1  49432  prcofdiag1  49521  prcofdiag  49522  oppfdiag1  49542  oppfdiag  49544  termcfuncval  49660