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Theorem homfeq 16567
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 2770 . . . . 5 (𝜑 → (𝑥𝐻𝑦) = (𝑥𝐻𝑦))
31, 1, 2mpt2eq123dv 6862 . . . 4 (𝜑 → (𝑥𝐵, 𝑦𝐵 ↦ (𝑥𝐻𝑦)) = (𝑥 ∈ (Base‘𝐶), 𝑦 ∈ (Base‘𝐶) ↦ (𝑥𝐻𝑦)))
4 eqid 2769 . . . . 5 (Homf𝐶) = (Homf𝐶)
5 eqid 2769 . . . . 5 (Base‘𝐶) = (Base‘𝐶)
6 homfeq.h . . . . 5 𝐻 = (Hom ‘𝐶)
74, 5, 6homffval 16563 . . . 4 (Homf𝐶) = (𝑥 ∈ (Base‘𝐶), 𝑦 ∈ (Base‘𝐶) ↦ (𝑥𝐻𝑦))
83, 7syl6reqr 2822 . . 3 (𝜑 → (Homf𝐶) = (𝑥𝐵, 𝑦𝐵 ↦ (𝑥𝐻𝑦)))
9 homfeq.2 . . . . 5 (𝜑𝐵 = (Base‘𝐷))
10 eqidd 2770 . . . . 5 (𝜑 → (𝑥𝐽𝑦) = (𝑥𝐽𝑦))
119, 9, 10mpt2eq123dv 6862 . . . 4 (𝜑 → (𝑥𝐵, 𝑦𝐵 ↦ (𝑥𝐽𝑦)) = (𝑥 ∈ (Base‘𝐷), 𝑦 ∈ (Base‘𝐷) ↦ (𝑥𝐽𝑦)))
12 eqid 2769 . . . . 5 (Homf𝐷) = (Homf𝐷)
13 eqid 2769 . . . . 5 (Base‘𝐷) = (Base‘𝐷)
14 homfeq.j . . . . 5 𝐽 = (Hom ‘𝐷)
1512, 13, 14homffval 16563 . . . 4 (Homf𝐷) = (𝑥 ∈ (Base‘𝐷), 𝑦 ∈ (Base‘𝐷) ↦ (𝑥𝐽𝑦))
1611, 15syl6reqr 2822 . . 3 (𝜑 → (Homf𝐷) = (𝑥𝐵, 𝑦𝐵 ↦ (𝑥𝐽𝑦)))
178, 16eqeq12d 2784 . 2 (𝜑 → ((Homf𝐶) = (Homf𝐷) ↔ (𝑥𝐵, 𝑦𝐵 ↦ (𝑥𝐻𝑦)) = (𝑥𝐵, 𝑦𝐵 ↦ (𝑥𝐽𝑦))))
18 ovex 6821 . . . 4 (𝑥𝐻𝑦) ∈ V
1918rgen2w 3072 . . 3 𝑥𝐵𝑦𝐵 (𝑥𝐻𝑦) ∈ V
20 mpt22eqb 6914 . . 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 1629  wcel 2143  wral 3059  Vcvv 3348  cfv 6030  (class class class)co 6791  cmpt2 6793  Basecbs 16070  Hom chom 16166  Homf chomf 16540
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1868  ax-4 1883  ax-5 1989  ax-6 2055  ax-7 2091  ax-8 2145  ax-9 2152  ax-10 2172  ax-11 2188  ax-12 2201  ax-13 2406  ax-ext 2749  ax-rep 4901  ax-sep 4911  ax-nul 4919  ax-pow 4970  ax-pr 5033  ax-un 7094
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1071  df-tru 1632  df-ex 1851  df-nf 1856  df-sb 2048  df-eu 2620  df-mo 2621  df-clab 2756  df-cleq 2762  df-clel 2765  df-nfc 2900  df-ne 2942  df-ral 3064  df-rex 3065  df-reu 3066  df-rab 3068  df-v 3350  df-sbc 3585  df-csb 3680  df-dif 3723  df-un 3725  df-in 3727  df-ss 3734  df-nul 4061  df-if 4223  df-pw 4296  df-sn 4314  df-pr 4316  df-op 4320  df-uni 4572  df-iun 4653  df-br 4784  df-opab 4844  df-mpt 4861  df-id 5156  df-xp 5254  df-rel 5255  df-cnv 5256  df-co 5257  df-dm 5258  df-rn 5259  df-res 5260  df-ima 5261  df-iota 5993  df-fun 6032  df-fn 6033  df-f 6034  df-f1 6035  df-fo 6036  df-f1o 6037  df-fv 6038  df-ov 6794  df-oprab 6795  df-mpt2 6796  df-1st 7313  df-2nd 7314  df-homf 16544
This theorem is referenced by:  homfeqd  16568  fullresc  16724  resssetc  16955  resscatc  16968
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