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Theorem hof1fval 16716
 Description: The object part of the Hom functor is the Homf operation, which is just a functionalized version of Hom. That is, it is a two argument function, which maps 𝑋, 𝑌 to the set of morphisms from 𝑋 to 𝑌. (Contributed by Mario Carneiro, 15-Jan-2017.)
Hypotheses
Ref Expression
hofval.m 𝑀 = (HomF𝐶)
hofval.c (𝜑𝐶 ∈ Cat)
Assertion
Ref Expression
hof1fval (𝜑 → (1st𝑀) = (Homf𝐶))

Proof of Theorem hof1fval
Dummy variables 𝑓 𝑔 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 hofval.m . . 3 𝑀 = (HomF𝐶)
2 hofval.c . . 3 (𝜑𝐶 ∈ Cat)
3 eqid 2610 . . 3 (Base‘𝐶) = (Base‘𝐶)
4 eqid 2610 . . 3 (Hom ‘𝐶) = (Hom ‘𝐶)
5 eqid 2610 . . 3 (comp‘𝐶) = (comp‘𝐶)
61, 2, 3, 4, 5hofval 16715 . 2 (𝜑𝑀 = ⟨(Homf𝐶), (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶)), 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐶)) ↦ (𝑓 ∈ ((1st𝑦)(Hom ‘𝐶)(1st𝑥)), 𝑔 ∈ ((2nd𝑥)(Hom ‘𝐶)(2nd𝑦)) ↦ ( ∈ ((Hom ‘𝐶)‘𝑥) ↦ ((𝑔(𝑥(comp‘𝐶)(2nd𝑦)))(⟨(1st𝑦), (1st𝑥)⟩(comp‘𝐶)(2nd𝑦))𝑓))))⟩)
7 fvex 6113 . . 3 (Homf𝐶) ∈ V
8 fvex 6113 . . . . 5 (Base‘𝐶) ∈ V
98, 8xpex 6860 . . . 4 ((Base‘𝐶) × (Base‘𝐶)) ∈ V
109, 9mpt2ex 7136 . . 3 (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶)), 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐶)) ↦ (𝑓 ∈ ((1st𝑦)(Hom ‘𝐶)(1st𝑥)), 𝑔 ∈ ((2nd𝑥)(Hom ‘𝐶)(2nd𝑦)) ↦ ( ∈ ((Hom ‘𝐶)‘𝑥) ↦ ((𝑔(𝑥(comp‘𝐶)(2nd𝑦)))(⟨(1st𝑦), (1st𝑥)⟩(comp‘𝐶)(2nd𝑦))𝑓)))) ∈ V
117, 10op1std 7069 . 2 (𝑀 = ⟨(Homf𝐶), (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶)), 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐶)) ↦ (𝑓 ∈ ((1st𝑦)(Hom ‘𝐶)(1st𝑥)), 𝑔 ∈ ((2nd𝑥)(Hom ‘𝐶)(2nd𝑦)) ↦ ( ∈ ((Hom ‘𝐶)‘𝑥) ↦ ((𝑔(𝑥(comp‘𝐶)(2nd𝑦)))(⟨(1st𝑦), (1st𝑥)⟩(comp‘𝐶)(2nd𝑦))𝑓))))⟩ → (1st𝑀) = (Homf𝐶))
126, 11syl 17 1 (𝜑 → (1st𝑀) = (Homf𝐶))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1475   ∈ wcel 1977  ⟨cop 4131   ↦ cmpt 4643   × cxp 5036  ‘cfv 5804  (class class class)co 6549   ↦ cmpt2 6551  1st c1st 7057  2nd c2nd 7058  Basecbs 15695  Hom chom 15779  compcco 15780  Catccat 16148  Homf chomf 16150  HomFchof 16711 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-1st 7059  df-2nd 7060  df-hof 16713 This theorem is referenced by:  hof1  16717
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