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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 2611 | . . . . 5 ⊢ (𝜑 → (𝑥𝐻𝑦) = (𝑥𝐻𝑦)) | |
3 | 1, 1, 2 | mpt2eq123dv 6615 | . . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥𝐻𝑦)) = (𝑥 ∈ (Base‘𝐶), 𝑦 ∈ (Base‘𝐶) ↦ (𝑥𝐻𝑦))) |
4 | eqid 2610 | . . . . 5 ⊢ (Homf ‘𝐶) = (Homf ‘𝐶) | |
5 | eqid 2610 | . . . . 5 ⊢ (Base‘𝐶) = (Base‘𝐶) | |
6 | homfeq.h | . . . . 5 ⊢ 𝐻 = (Hom ‘𝐶) | |
7 | 4, 5, 6 | homffval 16173 | . . . 4 ⊢ (Homf ‘𝐶) = (𝑥 ∈ (Base‘𝐶), 𝑦 ∈ (Base‘𝐶) ↦ (𝑥𝐻𝑦)) |
8 | 3, 7 | syl6reqr 2663 | . . 3 ⊢ (𝜑 → (Homf ‘𝐶) = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥𝐻𝑦))) |
9 | homfeq.2 | . . . . 5 ⊢ (𝜑 → 𝐵 = (Base‘𝐷)) | |
10 | eqidd 2611 | . . . . 5 ⊢ (𝜑 → (𝑥𝐽𝑦) = (𝑥𝐽𝑦)) | |
11 | 9, 9, 10 | mpt2eq123dv 6615 | . . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥𝐽𝑦)) = (𝑥 ∈ (Base‘𝐷), 𝑦 ∈ (Base‘𝐷) ↦ (𝑥𝐽𝑦))) |
12 | eqid 2610 | . . . . 5 ⊢ (Homf ‘𝐷) = (Homf ‘𝐷) | |
13 | eqid 2610 | . . . . 5 ⊢ (Base‘𝐷) = (Base‘𝐷) | |
14 | homfeq.j | . . . . 5 ⊢ 𝐽 = (Hom ‘𝐷) | |
15 | 12, 13, 14 | homffval 16173 | . . . 4 ⊢ (Homf ‘𝐷) = (𝑥 ∈ (Base‘𝐷), 𝑦 ∈ (Base‘𝐷) ↦ (𝑥𝐽𝑦)) |
16 | 11, 15 | syl6reqr 2663 | . . 3 ⊢ (𝜑 → (Homf ‘𝐷) = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥𝐽𝑦))) |
17 | 8, 16 | eqeq12d 2625 | . 2 ⊢ (𝜑 → ((Homf ‘𝐶) = (Homf ‘𝐷) ↔ (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥𝐻𝑦)) = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥𝐽𝑦)))) |
18 | ovex 6577 | . . . 4 ⊢ (𝑥𝐻𝑦) ∈ V | |
19 | 18 | rgen2w 2909 | . . 3 ⊢ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥𝐻𝑦) ∈ V |
20 | mpt22eqb 6667 | . . 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 275 | 1 ⊢ (𝜑 → ((Homf ‘𝐶) = (Homf ‘𝐷) ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥𝐻𝑦) = (𝑥𝐽𝑦))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 = wceq 1475 ∈ wcel 1977 ∀wral 2896 Vcvv 3173 ‘cfv 5804 (class class class)co 6549 ↦ cmpt2 6551 Basecbs 15695 Hom chom 15779 Homf chomf 16150 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-1st 7059 df-2nd 7060 df-homf 16154 |
This theorem is referenced by: homfeqd 16178 fullresc 16334 resssetc 16565 resscatc 16578 |
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