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| 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 2736 | . . . . 5 ⊢ (Homf ‘𝐶) = (Homf ‘𝐶) | |
| 2 | eqid 2736 | . . . . 5 ⊢ (Base‘𝐶) = (Base‘𝐶) | |
| 3 | homfeq.h | . . . . 5 ⊢ 𝐻 = (Hom ‘𝐶) | |
| 4 | 1, 2, 3 | homffval 17734 | . . . 4 ⊢ (Homf ‘𝐶) = (𝑥 ∈ (Base‘𝐶), 𝑦 ∈ (Base‘𝐶) ↦ (𝑥𝐻𝑦)) | 
| 5 | homfeq.1 | . . . . 5 ⊢ (𝜑 → 𝐵 = (Base‘𝐶)) | |
| 6 | eqidd 2737 | . . . . 5 ⊢ (𝜑 → (𝑥𝐻𝑦) = (𝑥𝐻𝑦)) | |
| 7 | 5, 5, 6 | mpoeq123dv 7509 | . . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥𝐻𝑦)) = (𝑥 ∈ (Base‘𝐶), 𝑦 ∈ (Base‘𝐶) ↦ (𝑥𝐻𝑦))) | 
| 8 | 4, 7 | eqtr4id 2795 | . . 3 ⊢ (𝜑 → (Homf ‘𝐶) = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥𝐻𝑦))) | 
| 9 | eqid 2736 | . . . . 5 ⊢ (Homf ‘𝐷) = (Homf ‘𝐷) | |
| 10 | eqid 2736 | . . . . 5 ⊢ (Base‘𝐷) = (Base‘𝐷) | |
| 11 | homfeq.j | . . . . 5 ⊢ 𝐽 = (Hom ‘𝐷) | |
| 12 | 9, 10, 11 | homffval 17734 | . . . 4 ⊢ (Homf ‘𝐷) = (𝑥 ∈ (Base‘𝐷), 𝑦 ∈ (Base‘𝐷) ↦ (𝑥𝐽𝑦)) | 
| 13 | homfeq.2 | . . . . 5 ⊢ (𝜑 → 𝐵 = (Base‘𝐷)) | |
| 14 | eqidd 2737 | . . . . 5 ⊢ (𝜑 → (𝑥𝐽𝑦) = (𝑥𝐽𝑦)) | |
| 15 | 13, 13, 14 | mpoeq123dv 7509 | . . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥𝐽𝑦)) = (𝑥 ∈ (Base‘𝐷), 𝑦 ∈ (Base‘𝐷) ↦ (𝑥𝐽𝑦))) | 
| 16 | 12, 15 | eqtr4id 2795 | . . 3 ⊢ (𝜑 → (Homf ‘𝐷) = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥𝐽𝑦))) | 
| 17 | 8, 16 | eqeq12d 2752 | . 2 ⊢ (𝜑 → ((Homf ‘𝐶) = (Homf ‘𝐷) ↔ (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥𝐻𝑦)) = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥𝐽𝑦)))) | 
| 18 | ovex 7465 | . . . 4 ⊢ (𝑥𝐻𝑦) ∈ V | |
| 19 | 18 | rgen2w 3065 | . . 3 ⊢ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥𝐻𝑦) ∈ V | 
| 20 | mpo2eqb 7566 | . . 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 1539 ∈ wcel 2107 ∀wral 3060 Vcvv 3479 ‘cfv 6560 (class class class)co 7432 ∈ cmpo 7434 Basecbs 17248 Hom chom 17309 Homf chomf 17710 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-rep 5278 ax-sep 5295 ax-nul 5305 ax-pow 5364 ax-pr 5431 ax-un 7756 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-ral 3061 df-rex 3070 df-reu 3380 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-iun 4992 df-br 5143 df-opab 5205 df-mpt 5225 df-id 5577 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-iota 6513 df-fun 6562 df-fn 6563 df-f 6564 df-f1 6565 df-fo 6566 df-f1o 6567 df-fv 6568 df-ov 7435 df-oprab 7436 df-mpo 7437 df-1st 8015 df-2nd 8016 df-homf 17714 | 
| This theorem is referenced by: homfeqd 17739 fullresc 17897 resssetc 18138 resscatc 18155 | 
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