Theorem funcfn2 17831

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 2729	. . 3 ⊢ (Hom ‘𝐷) = (Hom ‘𝐷)
3		eqid 2729	. . 3 ⊢ (Hom ‘𝐸) = (Hom ‘𝐸)
4		funcfn2.f	. . 3 ⊢ (𝜑 → 𝐹(𝐷 Func 𝐸)𝐺)
5	1, 2, 3, 4	funcixp 17829	. 2 ⊢ (𝜑 → 𝐺 ∈ X𝑥 ∈ (𝐵 × 𝐵)(((𝐹‘(1st ‘𝑥))(Hom ‘𝐸)(𝐹‘(2nd ‘𝑥))) ↑m ((Hom ‘𝐷)‘𝑥)))
6		ixpfn 8876	. 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 1540   ∈ wcel 2109   class class class wbr 5107   × cxp 5636   Fn wfn 6506  ‘cfv 6511  (class class class)co 7387  1^st c1st 7966  2^nd c2nd 7967   ↑_m cmap 8799  Xcixp 8870  Basecbs 17179  Hom chom 17231   Func cfunc 17816

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 2701  ax-rep 5234  ax-sep 5251  ax-nul 5261  ax-pow 5320  ax-pr 5387  ax-un 7711

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 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3406  df-v 3449  df-sbc 3754  df-csb 3863  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-iun 4957  df-br 5108  df-opab 5170  df-mpt 5189  df-id 5533  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-iota 6464  df-fun 6513  df-fn 6514  df-f 6515  df-fv 6519  df-ov 7390  df-oprab 7391  df-mpo 7392  df-map 8801  df-ixp 8871  df-func 17820

This theorem is referenced by:  funcoppc  17837  cofuval  17844  cofulid  17852  cofurid  17853  prf1st  18165  prf2nd  18166  1st2ndprf  18167  curfuncf  18199  uncfcurf  18200  curf2ndf  18208  oppfvallem  49124  uptposlem  49186  diag1  49293  prcofdiag1  49382  prcofdiag  49383  oppfdiag1  49403  oppfdiag  49405  termcfuncval  49521