Theorem funcfn2 17927

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 2763	. . 3 ⊢ (Hom ‘𝐷) = (Hom ‘𝐷)
3		eqid 2763	. . 3 ⊢ (Hom ‘𝐸) = (Hom ‘𝐸)
4		funcfn2.f	. . 3 ⊢ (𝜑 → 𝐹(𝐷 Func 𝐸)𝐺)
5	1, 2, 3, 4	funcixp 17925	. 2 ⊢ (𝜑 → 𝐺 ∈ X𝑥 ∈ (𝐵 × 𝐵)(((𝐹‘(1st ‘𝑥))(Hom ‘𝐸)(𝐹‘(2nd ‘𝑥))) ↑m ((Hom ‘𝐷)‘𝑥)))
6		ixpfn 8902	. 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 18	1 ⊢ (𝜑 → 𝐺 Fn (𝐵 × 𝐵))

Syntax hints:   → wi 4   = wceq 1570   ∈ wcel 2143   class class class wbr 5110   × cxp 5661   Fn wfn 6533  ‘cfv 6538  (class class class)co 7412  1^st c1st 7985  2^nd c2nd 7986   ↑_m cmap 8825  Xcixp 8896  Basecbs 17270  Hom chom 17322   Func cfunc 17912

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-rep 5239  ax-sep 5258  ax-nul 5270  ax-pow 5338  ax-pr 5406  ax-un 7734

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-rab 3417  df-v 3457  df-sbc 3746  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4288  df-if 4489  df-pw 4565  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4874  df-iun 4959  df-br 5111  df-opab 5175  df-mpt 5194  df-id 5558  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-fv 6546  df-ov 7415  df-oprab 7416  df-mpo 7417  df-map 8827  df-ixp 8897  df-func 17916

This theorem is referenced by:  funcoppc  17933  cofuval  17940  cofulid  17948  cofurid  17949  prf1st  18261  prf2nd  18262  1st2ndprf  18263  curfuncf  18295  uncfcurf  18296  curf2ndf  18304  oppfvallem  49896  uptposlem  49958  diag1  50065  prcofdiag1  50154  prcofdiag  50155  oppfdiag1  50175  oppfdiag  50177  termcfuncval  50293