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Theorem hof1fval 18278
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 2726 . . 3 (Base‘𝐶) = (Base‘𝐶)
4 eqid 2726 . . 3 (Hom ‘𝐶) = (Hom ‘𝐶)
5 eqid 2726 . . 3 (comp‘𝐶) = (comp‘𝐶)
61, 2, 3, 4, 5hofval 18277 . 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 6914 . . 3 (Homf𝐶) ∈ V
8 fvex 6914 . . . . 5 (Base‘𝐶) ∈ V
98, 8xpex 7761 . . . 4 ((Base‘𝐶) × (Base‘𝐶)) ∈ V
109, 9mpoex 8093 . . 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 8013 . 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 1534  wcel 2099  cop 4639  cmpt 5236   × cxp 5680  cfv 6554  (class class class)co 7424  cmpo 7426  1st c1st 8001  2nd c2nd 8002  Basecbs 17213  Hom chom 17277  compcco 17278  Catccat 17677  Homf chomf 17679  HomFchof 18273
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-rep 5290  ax-sep 5304  ax-nul 5311  ax-pow 5369  ax-pr 5433  ax-un 7746
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ne 2931  df-ral 3052  df-rex 3061  df-reu 3365  df-rab 3420  df-v 3464  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4326  df-if 4534  df-pw 4609  df-sn 4634  df-pr 4636  df-op 4640  df-uni 4914  df-iun 5003  df-br 5154  df-opab 5216  df-mpt 5237  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-iota 6506  df-fun 6556  df-fn 6557  df-f 6558  df-f1 6559  df-fo 6560  df-f1o 6561  df-fv 6562  df-ov 7427  df-oprab 7428  df-mpo 7429  df-1st 8003  df-2nd 8004  df-hof 18275
This theorem is referenced by:  hof1  18279
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