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Theorem funcfn2 16751
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 2761 . . 3 (Hom ‘𝐷) = (Hom ‘𝐷)
3 eqid 2761 . . 3 (Hom ‘𝐸) = (Hom ‘𝐸)
4 funcfn2.f . . 3 (𝜑𝐹(𝐷 Func 𝐸)𝐺)
51, 2, 3, 4funcixp 16749 . 2 (𝜑𝐺X𝑥 ∈ (𝐵 × 𝐵)(((𝐹‘(1st𝑥))(Hom ‘𝐸)(𝐹‘(2nd𝑥))) ↑𝑚 ((Hom ‘𝐷)‘𝑥)))
6 ixpfn 8083 . 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 1632  wcel 2140   class class class wbr 4805   × cxp 5265   Fn wfn 6045  cfv 6050  (class class class)co 6815  1st c1st 7333  2nd c2nd 7334  𝑚 cmap 8026  Xcixp 8077  Basecbs 16080  Hom chom 16175   Func cfunc 16736
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1989  ax-6 2055  ax-7 2091  ax-8 2142  ax-9 2149  ax-10 2169  ax-11 2184  ax-12 2197  ax-13 2392  ax-ext 2741  ax-rep 4924  ax-sep 4934  ax-nul 4942  ax-pow 4993  ax-pr 5056  ax-un 7116
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2048  df-eu 2612  df-mo 2613  df-clab 2748  df-cleq 2754  df-clel 2757  df-nfc 2892  df-ne 2934  df-ral 3056  df-rex 3057  df-reu 3058  df-rab 3060  df-v 3343  df-sbc 3578  df-csb 3676  df-dif 3719  df-un 3721  df-in 3723  df-ss 3730  df-nul 4060  df-if 4232  df-pw 4305  df-sn 4323  df-pr 4325  df-op 4329  df-uni 4590  df-iun 4675  df-br 4806  df-opab 4866  df-mpt 4883  df-id 5175  df-xp 5273  df-rel 5274  df-cnv 5275  df-co 5276  df-dm 5277  df-rn 5278  df-res 5279  df-ima 5280  df-iota 6013  df-fun 6052  df-fn 6053  df-f 6054  df-f1 6055  df-fo 6056  df-f1o 6057  df-fv 6058  df-ov 6818  df-oprab 6819  df-mpt2 6820  df-map 8028  df-ixp 8078  df-func 16740
This theorem is referenced by:  funcoppc  16757  cofuval  16764  cofulid  16772  cofurid  16773  prf1st  17066  prf2nd  17067  1st2ndprf  17068  curfuncf  17100  uncfcurf  17101  curf2ndf  17109
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