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Theorem homfeq 16964
Description: Condition for two categories with the same base to have the same hom-sets. (Contributed by Mario Carneiro, 6-Jan-2017.)
Hypotheses
Ref Expression
homfeq.h 𝐻 = (Hom ‘𝐶)
homfeq.j 𝐽 = (Hom ‘𝐷)
homfeq.1 (𝜑𝐵 = (Base‘𝐶))
homfeq.2 (𝜑𝐵 = (Base‘𝐷))
Assertion
Ref Expression
homfeq (𝜑 → ((Homf𝐶) = (Homf𝐷) ↔ ∀𝑥𝐵𝑦𝐵 (𝑥𝐻𝑦) = (𝑥𝐽𝑦)))
Distinct variable groups:   𝑥,𝑦,𝐵   𝑥,𝐶,𝑦   𝑥,𝐷,𝑦   𝑥,𝐻,𝑦   𝜑,𝑥,𝑦   𝑥,𝐽,𝑦

Proof of Theorem homfeq
StepHypRef Expression
1 homfeq.1 . . . . 5 (𝜑𝐵 = (Base‘𝐶))
2 eqidd 2825 . . . . 5 (𝜑 → (𝑥𝐻𝑦) = (𝑥𝐻𝑦))
31, 1, 2mpoeq123dv 7222 . . . 4 (𝜑 → (𝑥𝐵, 𝑦𝐵 ↦ (𝑥𝐻𝑦)) = (𝑥 ∈ (Base‘𝐶), 𝑦 ∈ (Base‘𝐶) ↦ (𝑥𝐻𝑦)))
4 eqid 2824 . . . . 5 (Homf𝐶) = (Homf𝐶)
5 eqid 2824 . . . . 5 (Base‘𝐶) = (Base‘𝐶)
6 homfeq.h . . . . 5 𝐻 = (Hom ‘𝐶)
74, 5, 6homffval 16960 . . . 4 (Homf𝐶) = (𝑥 ∈ (Base‘𝐶), 𝑦 ∈ (Base‘𝐶) ↦ (𝑥𝐻𝑦))
83, 7syl6reqr 2878 . . 3 (𝜑 → (Homf𝐶) = (𝑥𝐵, 𝑦𝐵 ↦ (𝑥𝐻𝑦)))
9 homfeq.2 . . . . 5 (𝜑𝐵 = (Base‘𝐷))
10 eqidd 2825 . . . . 5 (𝜑 → (𝑥𝐽𝑦) = (𝑥𝐽𝑦))
119, 9, 10mpoeq123dv 7222 . . . 4 (𝜑 → (𝑥𝐵, 𝑦𝐵 ↦ (𝑥𝐽𝑦)) = (𝑥 ∈ (Base‘𝐷), 𝑦 ∈ (Base‘𝐷) ↦ (𝑥𝐽𝑦)))
12 eqid 2824 . . . . 5 (Homf𝐷) = (Homf𝐷)
13 eqid 2824 . . . . 5 (Base‘𝐷) = (Base‘𝐷)
14 homfeq.j . . . . 5 𝐽 = (Hom ‘𝐷)
1512, 13, 14homffval 16960 . . . 4 (Homf𝐷) = (𝑥 ∈ (Base‘𝐷), 𝑦 ∈ (Base‘𝐷) ↦ (𝑥𝐽𝑦))
1611, 15syl6reqr 2878 . . 3 (𝜑 → (Homf𝐷) = (𝑥𝐵, 𝑦𝐵 ↦ (𝑥𝐽𝑦)))
178, 16eqeq12d 2840 . 2 (𝜑 → ((Homf𝐶) = (Homf𝐷) ↔ (𝑥𝐵, 𝑦𝐵 ↦ (𝑥𝐻𝑦)) = (𝑥𝐵, 𝑦𝐵 ↦ (𝑥𝐽𝑦))))
18 ovex 7182 . . . 4 (𝑥𝐻𝑦) ∈ V
1918rgen2w 3146 . . 3 𝑥𝐵𝑦𝐵 (𝑥𝐻𝑦) ∈ V
20 mpo2eqb 7276 . . 3 (∀𝑥𝐵𝑦𝐵 (𝑥𝐻𝑦) ∈ V → ((𝑥𝐵, 𝑦𝐵 ↦ (𝑥𝐻𝑦)) = (𝑥𝐵, 𝑦𝐵 ↦ (𝑥𝐽𝑦)) ↔ ∀𝑥𝐵𝑦𝐵 (𝑥𝐻𝑦) = (𝑥𝐽𝑦)))
2119, 20ax-mp 5 . 2 ((𝑥𝐵, 𝑦𝐵 ↦ (𝑥𝐻𝑦)) = (𝑥𝐵, 𝑦𝐵 ↦ (𝑥𝐽𝑦)) ↔ ∀𝑥𝐵𝑦𝐵 (𝑥𝐻𝑦) = (𝑥𝐽𝑦))
2217, 21syl6bb 290 1 (𝜑 → ((Homf𝐶) = (Homf𝐷) ↔ ∀𝑥𝐵𝑦𝐵 (𝑥𝐻𝑦) = (𝑥𝐽𝑦)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209   = wceq 1538  wcel 2115  wral 3133  Vcvv 3480  cfv 6343  (class class class)co 7149  cmpo 7151  Basecbs 16483  Hom chom 16576  Homf chomf 16937
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-1st 7684  df-2nd 7685  df-homf 16941
This theorem is referenced by:  homfeqd  16965  fullresc  17121  resssetc  17352  resscatc  17365
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