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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 2735 | . . . . 5 ⊢ (Homf ‘𝐶) = (Homf ‘𝐶) | |
2 | eqid 2735 | . . . . 5 ⊢ (Base‘𝐶) = (Base‘𝐶) | |
3 | homfeq.h | . . . . 5 ⊢ 𝐻 = (Hom ‘𝐶) | |
4 | 1, 2, 3 | homffval 17735 | . . . 4 ⊢ (Homf ‘𝐶) = (𝑥 ∈ (Base‘𝐶), 𝑦 ∈ (Base‘𝐶) ↦ (𝑥𝐻𝑦)) |
5 | homfeq.1 | . . . . 5 ⊢ (𝜑 → 𝐵 = (Base‘𝐶)) | |
6 | eqidd 2736 | . . . . 5 ⊢ (𝜑 → (𝑥𝐻𝑦) = (𝑥𝐻𝑦)) | |
7 | 5, 5, 6 | mpoeq123dv 7508 | . . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥𝐻𝑦)) = (𝑥 ∈ (Base‘𝐶), 𝑦 ∈ (Base‘𝐶) ↦ (𝑥𝐻𝑦))) |
8 | 4, 7 | eqtr4id 2794 | . . 3 ⊢ (𝜑 → (Homf ‘𝐶) = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥𝐻𝑦))) |
9 | eqid 2735 | . . . . 5 ⊢ (Homf ‘𝐷) = (Homf ‘𝐷) | |
10 | eqid 2735 | . . . . 5 ⊢ (Base‘𝐷) = (Base‘𝐷) | |
11 | homfeq.j | . . . . 5 ⊢ 𝐽 = (Hom ‘𝐷) | |
12 | 9, 10, 11 | homffval 17735 | . . . 4 ⊢ (Homf ‘𝐷) = (𝑥 ∈ (Base‘𝐷), 𝑦 ∈ (Base‘𝐷) ↦ (𝑥𝐽𝑦)) |
13 | homfeq.2 | . . . . 5 ⊢ (𝜑 → 𝐵 = (Base‘𝐷)) | |
14 | eqidd 2736 | . . . . 5 ⊢ (𝜑 → (𝑥𝐽𝑦) = (𝑥𝐽𝑦)) | |
15 | 13, 13, 14 | mpoeq123dv 7508 | . . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥𝐽𝑦)) = (𝑥 ∈ (Base‘𝐷), 𝑦 ∈ (Base‘𝐷) ↦ (𝑥𝐽𝑦))) |
16 | 12, 15 | eqtr4id 2794 | . . 3 ⊢ (𝜑 → (Homf ‘𝐷) = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥𝐽𝑦))) |
17 | 8, 16 | eqeq12d 2751 | . 2 ⊢ (𝜑 → ((Homf ‘𝐶) = (Homf ‘𝐷) ↔ (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥𝐻𝑦)) = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥𝐽𝑦)))) |
18 | ovex 7464 | . . . 4 ⊢ (𝑥𝐻𝑦) ∈ V | |
19 | 18 | rgen2w 3064 | . . 3 ⊢ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥𝐻𝑦) ∈ V |
20 | mpo2eqb 7565 | . . 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 287 | 1 ⊢ (𝜑 → ((Homf ‘𝐶) = (Homf ‘𝐷) ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥𝐻𝑦) = (𝑥𝐽𝑦))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 = wceq 1537 ∈ wcel 2106 ∀wral 3059 Vcvv 3478 ‘cfv 6563 (class class class)co 7431 ∈ cmpo 7433 Basecbs 17245 Hom chom 17309 Homf chomf 17711 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-ov 7434 df-oprab 7435 df-mpo 7436 df-1st 8013 df-2nd 8014 df-homf 17715 |
This theorem is referenced by: homfeqd 17740 fullresc 17902 resssetc 18146 resscatc 18163 |
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