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Theorem funcfn2 17915
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 (𝜑𝐹(𝐷 Func 𝐸)𝐺)
Assertion
Ref Expression
funcfn2 (𝜑𝐺 Fn (𝐵 × 𝐵))

Proof of Theorem funcfn2
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef 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 (𝜑𝐹(𝐷 Func 𝐸)𝐺)
51, 2, 3, 4funcixp 17913 . 2 (𝜑𝐺X𝑥 ∈ (𝐵 × 𝐵)(((𝐹‘(1st𝑥))(Hom ‘𝐸)(𝐹‘(2nd𝑥))) ↑m ((Hom ‘𝐷)‘𝑥)))
6 ixpfn 8944 . 2 (𝐺X𝑥 ∈ (𝐵 × 𝐵)(((𝐹‘(1st𝑥))(Hom ‘𝐸)(𝐹‘(2nd𝑥))) ↑m ((Hom ‘𝐷)‘𝑥)) → 𝐺 Fn (𝐵 × 𝐵))
75, 6syl 17 1 (𝜑𝐺 Fn (𝐵 × 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1539  wcel 2107   class class class wbr 5142   × cxp 5682   Fn wfn 6555  cfv 6560  (class class class)co 7432  1st c1st 8013  2nd c2nd 8014  m cmap 8867  Xcixp 8938  Basecbs 17248  Hom chom 17309   Func cfunc 17900
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-rep 5278  ax-sep 5295  ax-nul 5305  ax-pow 5364  ax-pr 5431  ax-un 7756
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-iun 4992  df-br 5143  df-opab 5205  df-mpt 5225  df-id 5577  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-fv 6568  df-ov 7435  df-oprab 7436  df-mpo 7437  df-map 8869  df-ixp 8939  df-func 17904
This theorem is referenced by:  funcoppc  17921  cofuval  17928  cofulid  17936  cofurid  17937  prf1st  18250  prf2nd  18251  1st2ndprf  18252  curfuncf  18284  uncfcurf  18285  curf2ndf  18293  diag1  49022  termcfuncval  49190
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