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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 | homfeq.1 | . . . . 5 ⊢ (𝜑 → 𝐵 = (Base‘𝐶)) | |
2 | eqidd 2822 | . . . . 5 ⊢ (𝜑 → (𝑥𝐻𝑦) = (𝑥𝐻𝑦)) | |
3 | 1, 1, 2 | mpoeq123dv 7229 | . . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥𝐻𝑦)) = (𝑥 ∈ (Base‘𝐶), 𝑦 ∈ (Base‘𝐶) ↦ (𝑥𝐻𝑦))) |
4 | eqid 2821 | . . . . 5 ⊢ (Homf ‘𝐶) = (Homf ‘𝐶) | |
5 | eqid 2821 | . . . . 5 ⊢ (Base‘𝐶) = (Base‘𝐶) | |
6 | homfeq.h | . . . . 5 ⊢ 𝐻 = (Hom ‘𝐶) | |
7 | 4, 5, 6 | homffval 16960 | . . . 4 ⊢ (Homf ‘𝐶) = (𝑥 ∈ (Base‘𝐶), 𝑦 ∈ (Base‘𝐶) ↦ (𝑥𝐻𝑦)) |
8 | 3, 7 | syl6reqr 2875 | . . 3 ⊢ (𝜑 → (Homf ‘𝐶) = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥𝐻𝑦))) |
9 | homfeq.2 | . . . . 5 ⊢ (𝜑 → 𝐵 = (Base‘𝐷)) | |
10 | eqidd 2822 | . . . . 5 ⊢ (𝜑 → (𝑥𝐽𝑦) = (𝑥𝐽𝑦)) | |
11 | 9, 9, 10 | mpoeq123dv 7229 | . . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥𝐽𝑦)) = (𝑥 ∈ (Base‘𝐷), 𝑦 ∈ (Base‘𝐷) ↦ (𝑥𝐽𝑦))) |
12 | eqid 2821 | . . . . 5 ⊢ (Homf ‘𝐷) = (Homf ‘𝐷) | |
13 | eqid 2821 | . . . . 5 ⊢ (Base‘𝐷) = (Base‘𝐷) | |
14 | homfeq.j | . . . . 5 ⊢ 𝐽 = (Hom ‘𝐷) | |
15 | 12, 13, 14 | homffval 16960 | . . . 4 ⊢ (Homf ‘𝐷) = (𝑥 ∈ (Base‘𝐷), 𝑦 ∈ (Base‘𝐷) ↦ (𝑥𝐽𝑦)) |
16 | 11, 15 | syl6reqr 2875 | . . 3 ⊢ (𝜑 → (Homf ‘𝐷) = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥𝐽𝑦))) |
17 | 8, 16 | eqeq12d 2837 | . 2 ⊢ (𝜑 → ((Homf ‘𝐶) = (Homf ‘𝐷) ↔ (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥𝐻𝑦)) = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥𝐽𝑦)))) |
18 | ovex 7189 | . . . 4 ⊢ (𝑥𝐻𝑦) ∈ V | |
19 | 18 | rgen2w 3151 | . . 3 ⊢ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥𝐻𝑦) ∈ V |
20 | mpo2eqb 7283 | . . 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 | syl6bb 289 | 1 ⊢ (𝜑 → ((Homf ‘𝐶) = (Homf ‘𝐷) ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥𝐻𝑦) = (𝑥𝐽𝑦))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 = wceq 1537 ∈ wcel 2114 ∀wral 3138 Vcvv 3494 ‘cfv 6355 (class class class)co 7156 ∈ cmpo 7158 Basecbs 16483 Hom chom 16576 Homf chomf 16937 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-id 5460 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-ov 7159 df-oprab 7160 df-mpo 7161 df-1st 7689 df-2nd 7690 df-homf 16941 |
This theorem is referenced by: homfeqd 16965 fullresc 17121 resssetc 17352 resscatc 17365 |
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