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Mirrors > Home > MPE Home > Th. List > homfeqd | Structured version Visualization version GIF version |
Description: If two structures have the same Hom slot, they have the same Hom-sets. (Contributed by Mario Carneiro, 4-Jan-2017.) |
Ref | Expression |
---|---|
homfeqd.1 | ⊢ (𝜑 → (Base‘𝐶) = (Base‘𝐷)) |
homfeqd.2 | ⊢ (𝜑 → (Hom ‘𝐶) = (Hom ‘𝐷)) |
Ref | Expression |
---|---|
homfeqd | ⊢ (𝜑 → (Homf ‘𝐶) = (Homf ‘𝐷)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | homfeqd.2 | . . . . 5 ⊢ (𝜑 → (Hom ‘𝐶) = (Hom ‘𝐷)) | |
2 | 1 | oveqd 7208 | . . . 4 ⊢ (𝜑 → (𝑥(Hom ‘𝐶)𝑦) = (𝑥(Hom ‘𝐷)𝑦)) |
3 | 2 | ralrimivw 3096 | . . 3 ⊢ (𝜑 → ∀𝑦 ∈ (Base‘𝐶)(𝑥(Hom ‘𝐶)𝑦) = (𝑥(Hom ‘𝐷)𝑦)) |
4 | 3 | ralrimivw 3096 | . 2 ⊢ (𝜑 → ∀𝑥 ∈ (Base‘𝐶)∀𝑦 ∈ (Base‘𝐶)(𝑥(Hom ‘𝐶)𝑦) = (𝑥(Hom ‘𝐷)𝑦)) |
5 | eqid 2736 | . . 3 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶) | |
6 | eqid 2736 | . . 3 ⊢ (Hom ‘𝐷) = (Hom ‘𝐷) | |
7 | eqidd 2737 | . . 3 ⊢ (𝜑 → (Base‘𝐶) = (Base‘𝐶)) | |
8 | homfeqd.1 | . . 3 ⊢ (𝜑 → (Base‘𝐶) = (Base‘𝐷)) | |
9 | 5, 6, 7, 8 | homfeq 17151 | . 2 ⊢ (𝜑 → ((Homf ‘𝐶) = (Homf ‘𝐷) ↔ ∀𝑥 ∈ (Base‘𝐶)∀𝑦 ∈ (Base‘𝐶)(𝑥(Hom ‘𝐶)𝑦) = (𝑥(Hom ‘𝐷)𝑦))) |
10 | 4, 9 | mpbird 260 | 1 ⊢ (𝜑 → (Homf ‘𝐶) = (Homf ‘𝐷)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1543 ∀wral 3051 ‘cfv 6358 (class class class)co 7191 Basecbs 16666 Hom chom 16760 Homf chomf 17123 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-rep 5164 ax-sep 5177 ax-nul 5184 ax-pow 5243 ax-pr 5307 ax-un 7501 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ne 2933 df-ral 3056 df-rex 3057 df-reu 3058 df-rab 3060 df-v 3400 df-sbc 3684 df-csb 3799 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-nul 4224 df-if 4426 df-pw 4501 df-sn 4528 df-pr 4530 df-op 4534 df-uni 4806 df-iun 4892 df-br 5040 df-opab 5102 df-mpt 5121 df-id 5440 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-rn 5547 df-res 5548 df-ima 5549 df-iota 6316 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-ov 7194 df-oprab 7195 df-mpo 7196 df-1st 7739 df-2nd 7740 df-homf 17127 |
This theorem is referenced by: (None) |
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