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Mirrors > Home > MPE Home > Th. List > homfeq | Structured version Visualization version GIF version |
Description: Condition for two categories with the same base to have the same hom-sets. (Contributed by Mario Carneiro, 6-Jan-2017.) |
Ref | Expression |
---|---|
homfeq.h | ⊢ 𝐻 = (Hom ‘𝐶) |
homfeq.j | ⊢ 𝐽 = (Hom ‘𝐷) |
homfeq.1 | ⊢ (𝜑 → 𝐵 = (Base‘𝐶)) |
homfeq.2 | ⊢ (𝜑 → 𝐵 = (Base‘𝐷)) |
Ref | Expression |
---|---|
homfeq | ⊢ (𝜑 → ((Homf ‘𝐶) = (Homf ‘𝐷) ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥𝐻𝑦) = (𝑥𝐽𝑦))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2737 | . . . . 5 ⊢ (Homf ‘𝐶) = (Homf ‘𝐶) | |
2 | eqid 2737 | . . . . 5 ⊢ (Base‘𝐶) = (Base‘𝐶) | |
3 | homfeq.h | . . . . 5 ⊢ 𝐻 = (Hom ‘𝐶) | |
4 | 1, 2, 3 | homffval 17530 | . . . 4 ⊢ (Homf ‘𝐶) = (𝑥 ∈ (Base‘𝐶), 𝑦 ∈ (Base‘𝐶) ↦ (𝑥𝐻𝑦)) |
5 | homfeq.1 | . . . . 5 ⊢ (𝜑 → 𝐵 = (Base‘𝐶)) | |
6 | eqidd 2738 | . . . . 5 ⊢ (𝜑 → (𝑥𝐻𝑦) = (𝑥𝐻𝑦)) | |
7 | 5, 5, 6 | mpoeq123dv 7426 | . . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥𝐻𝑦)) = (𝑥 ∈ (Base‘𝐶), 𝑦 ∈ (Base‘𝐶) ↦ (𝑥𝐻𝑦))) |
8 | 4, 7 | eqtr4id 2796 | . . 3 ⊢ (𝜑 → (Homf ‘𝐶) = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥𝐻𝑦))) |
9 | eqid 2737 | . . . . 5 ⊢ (Homf ‘𝐷) = (Homf ‘𝐷) | |
10 | eqid 2737 | . . . . 5 ⊢ (Base‘𝐷) = (Base‘𝐷) | |
11 | homfeq.j | . . . . 5 ⊢ 𝐽 = (Hom ‘𝐷) | |
12 | 9, 10, 11 | homffval 17530 | . . . 4 ⊢ (Homf ‘𝐷) = (𝑥 ∈ (Base‘𝐷), 𝑦 ∈ (Base‘𝐷) ↦ (𝑥𝐽𝑦)) |
13 | homfeq.2 | . . . . 5 ⊢ (𝜑 → 𝐵 = (Base‘𝐷)) | |
14 | eqidd 2738 | . . . . 5 ⊢ (𝜑 → (𝑥𝐽𝑦) = (𝑥𝐽𝑦)) | |
15 | 13, 13, 14 | mpoeq123dv 7426 | . . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥𝐽𝑦)) = (𝑥 ∈ (Base‘𝐷), 𝑦 ∈ (Base‘𝐷) ↦ (𝑥𝐽𝑦))) |
16 | 12, 15 | eqtr4id 2796 | . . 3 ⊢ (𝜑 → (Homf ‘𝐷) = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥𝐽𝑦))) |
17 | 8, 16 | eqeq12d 2753 | . 2 ⊢ (𝜑 → ((Homf ‘𝐶) = (Homf ‘𝐷) ↔ (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥𝐻𝑦)) = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥𝐽𝑦)))) |
18 | ovex 7384 | . . . 4 ⊢ (𝑥𝐻𝑦) ∈ V | |
19 | 18 | rgen2w 3067 | . . 3 ⊢ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥𝐻𝑦) ∈ V |
20 | mpo2eqb 7482 | . . 3 ⊢ (∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥𝐻𝑦) ∈ V → ((𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥𝐻𝑦)) = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥𝐽𝑦)) ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥𝐻𝑦) = (𝑥𝐽𝑦))) | |
21 | 19, 20 | ax-mp 5 | . 2 ⊢ ((𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥𝐻𝑦)) = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥𝐽𝑦)) ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥𝐻𝑦) = (𝑥𝐽𝑦)) |
22 | 17, 21 | bitrdi 286 | 1 ⊢ (𝜑 → ((Homf ‘𝐶) = (Homf ‘𝐷) ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (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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