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Theorem homfeq 16275
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 2622 . . . . 5 (𝜑 → (𝑥𝐻𝑦) = (𝑥𝐻𝑦))
31, 1, 2mpt2eq123dv 6670 . . . 4 (𝜑 → (𝑥𝐵, 𝑦𝐵 ↦ (𝑥𝐻𝑦)) = (𝑥 ∈ (Base‘𝐶), 𝑦 ∈ (Base‘𝐶) ↦ (𝑥𝐻𝑦)))
4 eqid 2621 . . . . 5 (Homf𝐶) = (Homf𝐶)
5 eqid 2621 . . . . 5 (Base‘𝐶) = (Base‘𝐶)
6 homfeq.h . . . . 5 𝐻 = (Hom ‘𝐶)
74, 5, 6homffval 16271 . . . 4 (Homf𝐶) = (𝑥 ∈ (Base‘𝐶), 𝑦 ∈ (Base‘𝐶) ↦ (𝑥𝐻𝑦))
83, 7syl6reqr 2674 . . 3 (𝜑 → (Homf𝐶) = (𝑥𝐵, 𝑦𝐵 ↦ (𝑥𝐻𝑦)))
9 homfeq.2 . . . . 5 (𝜑𝐵 = (Base‘𝐷))
10 eqidd 2622 . . . . 5 (𝜑 → (𝑥𝐽𝑦) = (𝑥𝐽𝑦))
119, 9, 10mpt2eq123dv 6670 . . . 4 (𝜑 → (𝑥𝐵, 𝑦𝐵 ↦ (𝑥𝐽𝑦)) = (𝑥 ∈ (Base‘𝐷), 𝑦 ∈ (Base‘𝐷) ↦ (𝑥𝐽𝑦)))
12 eqid 2621 . . . . 5 (Homf𝐷) = (Homf𝐷)
13 eqid 2621 . . . . 5 (Base‘𝐷) = (Base‘𝐷)
14 homfeq.j . . . . 5 𝐽 = (Hom ‘𝐷)
1512, 13, 14homffval 16271 . . . 4 (Homf𝐷) = (𝑥 ∈ (Base‘𝐷), 𝑦 ∈ (Base‘𝐷) ↦ (𝑥𝐽𝑦))
1611, 15syl6reqr 2674 . . 3 (𝜑 → (Homf𝐷) = (𝑥𝐵, 𝑦𝐵 ↦ (𝑥𝐽𝑦)))
178, 16eqeq12d 2636 . 2 (𝜑 → ((Homf𝐶) = (Homf𝐷) ↔ (𝑥𝐵, 𝑦𝐵 ↦ (𝑥𝐻𝑦)) = (𝑥𝐵, 𝑦𝐵 ↦ (𝑥𝐽𝑦))))
18 ovex 6632 . . . 4 (𝑥𝐻𝑦) ∈ V
1918rgen2w 2920 . . 3 𝑥𝐵𝑦𝐵 (𝑥𝐻𝑦) ∈ V
20 mpt22eqb 6722 . . 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 276 1 (𝜑 → ((Homf𝐶) = (Homf𝐷) ↔ ∀𝑥𝐵𝑦𝐵 (𝑥𝐻𝑦) = (𝑥𝐽𝑦)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196   = wceq 1480  wcel 1987  wral 2907  Vcvv 3186  cfv 5847  (class class class)co 6604  cmpt2 6606  Basecbs 15781  Hom chom 15873  Homf chomf 16248
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4731  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-1st 7113  df-2nd 7114  df-homf 16252
This theorem is referenced by:  homfeqd  16276  fullresc  16432  resssetc  16663  resscatc  16676
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