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Theorem homfeq 17534
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 eqid 2737 . . . . 5 (Homf𝐶) = (Homf𝐶)
2 eqid 2737 . . . . 5 (Base‘𝐶) = (Base‘𝐶)
3 homfeq.h . . . . 5 𝐻 = (Hom ‘𝐶)
41, 2, 3homffval 17530 . . . 4 (Homf𝐶) = (𝑥 ∈ (Base‘𝐶), 𝑦 ∈ (Base‘𝐶) ↦ (𝑥𝐻𝑦))
5 homfeq.1 . . . . 5 (𝜑𝐵 = (Base‘𝐶))
6 eqidd 2738 . . . . 5 (𝜑 → (𝑥𝐻𝑦) = (𝑥𝐻𝑦))
75, 5, 6mpoeq123dv 7426 . . . 4 (𝜑 → (𝑥𝐵, 𝑦𝐵 ↦ (𝑥𝐻𝑦)) = (𝑥 ∈ (Base‘𝐶), 𝑦 ∈ (Base‘𝐶) ↦ (𝑥𝐻𝑦)))
84, 7eqtr4id 2796 . . 3 (𝜑 → (Homf𝐶) = (𝑥𝐵, 𝑦𝐵 ↦ (𝑥𝐻𝑦)))
9 eqid 2737 . . . . 5 (Homf𝐷) = (Homf𝐷)
10 eqid 2737 . . . . 5 (Base‘𝐷) = (Base‘𝐷)
11 homfeq.j . . . . 5 𝐽 = (Hom ‘𝐷)
129, 10, 11homffval 17530 . . . 4 (Homf𝐷) = (𝑥 ∈ (Base‘𝐷), 𝑦 ∈ (Base‘𝐷) ↦ (𝑥𝐽𝑦))
13 homfeq.2 . . . . 5 (𝜑𝐵 = (Base‘𝐷))
14 eqidd 2738 . . . . 5 (𝜑 → (𝑥𝐽𝑦) = (𝑥𝐽𝑦))
1513, 13, 14mpoeq123dv 7426 . . . 4 (𝜑 → (𝑥𝐵, 𝑦𝐵 ↦ (𝑥𝐽𝑦)) = (𝑥 ∈ (Base‘𝐷), 𝑦 ∈ (Base‘𝐷) ↦ (𝑥𝐽𝑦)))
1612, 15eqtr4id 2796 . . 3 (𝜑 → (Homf𝐷) = (𝑥𝐵, 𝑦𝐵 ↦ (𝑥𝐽𝑦)))
178, 16eqeq12d 2753 . 2 (𝜑 → ((Homf𝐶) = (Homf𝐷) ↔ (𝑥𝐵, 𝑦𝐵 ↦ (𝑥𝐻𝑦)) = (𝑥𝐵, 𝑦𝐵 ↦ (𝑥𝐽𝑦))))
18 ovex 7384 . . . 4 (𝑥𝐻𝑦) ∈ V
1918rgen2w 3067 . . 3 𝑥𝐵𝑦𝐵 (𝑥𝐻𝑦) ∈ V
20 mpo2eqb 7482 . . 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, 21bitrdi 286 1 (𝜑 → ((Homf𝐶) = (Homf𝐷) ↔ ∀𝑥𝐵𝑦𝐵 (𝑥𝐻𝑦) = (𝑥𝐽𝑦)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205   = wceq 1541  wcel 2106  wral 3062  Vcvv 3443  cfv 6493  (class class class)co 7351  cmpo 7353  Basecbs 17043  Hom chom 17104  Homf chomf 17506
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2708  ax-rep 5240  ax-sep 5254  ax-nul 5261  ax-pow 5318  ax-pr 5382  ax-un 7664
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2887  df-ne 2942  df-ral 3063  df-rex 3072  df-reu 3352  df-rab 3406  df-v 3445  df-sbc 3738  df-csb 3854  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4281  df-if 4485  df-pw 4560  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4864  df-iun 4954  df-br 5104  df-opab 5166  df-mpt 5187  df-id 5529  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-iota 6445  df-fun 6495  df-fn 6496  df-f 6497  df-f1 6498  df-fo 6499  df-f1o 6500  df-fv 6501  df-ov 7354  df-oprab 7355  df-mpo 7356  df-1st 7913  df-2nd 7914  df-homf 17510
This theorem is referenced by:  homfeqd  17535  fullresc  17697  resssetc  17938  resscatc  17955
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