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Theorem funcfn2 17139
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 2824 . . 3 (Hom ‘𝐷) = (Hom ‘𝐷)
3 eqid 2824 . . 3 (Hom ‘𝐸) = (Hom ‘𝐸)
4 funcfn2.f . . 3 (𝜑𝐹(𝐷 Func 𝐸)𝐺)
51, 2, 3, 4funcixp 17137 . 2 (𝜑𝐺X𝑥 ∈ (𝐵 × 𝐵)(((𝐹‘(1st𝑥))(Hom ‘𝐸)(𝐹‘(2nd𝑥))) ↑m ((Hom ‘𝐷)‘𝑥)))
6 ixpfn 8463 . 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 1538  wcel 2115   class class class wbr 5052   × cxp 5540   Fn wfn 6338  cfv 6343  (class class class)co 7149  1st c1st 7682  2nd c2nd 7683  m cmap 8402  Xcixp 8457  Basecbs 16483  Hom chom 16576   Func cfunc 17124
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-rep 5176  ax-sep 5189  ax-nul 5196  ax-pow 5253  ax-pr 5317  ax-un 7455
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-ral 3138  df-rex 3139  df-reu 3140  df-rab 3142  df-v 3482  df-sbc 3759  df-csb 3867  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-if 4451  df-pw 4524  df-sn 4551  df-pr 4553  df-op 4557  df-uni 4825  df-iun 4907  df-br 5053  df-opab 5115  df-mpt 5133  df-id 5447  df-xp 5548  df-rel 5549  df-cnv 5550  df-co 5551  df-dm 5552  df-rn 5553  df-res 5554  df-ima 5555  df-iota 6302  df-fun 6345  df-fn 6346  df-f 6347  df-f1 6348  df-fo 6349  df-f1o 6350  df-fv 6351  df-ov 7152  df-oprab 7153  df-mpo 7154  df-map 8404  df-ixp 8458  df-func 17128
This theorem is referenced by:  funcoppc  17145  cofuval  17152  cofulid  17160  cofurid  17161  prf1st  17454  prf2nd  17455  1st2ndprf  17456  curfuncf  17488  uncfcurf  17489  curf2ndf  17497
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