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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 7167 | . . . 4 ⊢ (𝜑 → (𝑥(Hom ‘𝐶)𝑦) = (𝑥(Hom ‘𝐷)𝑦)) |
3 | 2 | ralrimivw 3183 | . . 3 ⊢ (𝜑 → ∀𝑦 ∈ (Base‘𝐶)(𝑥(Hom ‘𝐶)𝑦) = (𝑥(Hom ‘𝐷)𝑦)) |
4 | 3 | ralrimivw 3183 | . 2 ⊢ (𝜑 → ∀𝑥 ∈ (Base‘𝐶)∀𝑦 ∈ (Base‘𝐶)(𝑥(Hom ‘𝐶)𝑦) = (𝑥(Hom ‘𝐷)𝑦)) |
5 | eqid 2821 | . . 3 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶) | |
6 | eqid 2821 | . . 3 ⊢ (Hom ‘𝐷) = (Hom ‘𝐷) | |
7 | eqidd 2822 | . . 3 ⊢ (𝜑 → (Base‘𝐶) = (Base‘𝐶)) | |
8 | homfeqd.1 | . . 3 ⊢ (𝜑 → (Base‘𝐶) = (Base‘𝐷)) | |
9 | 5, 6, 7, 8 | homfeq 16958 | . 2 ⊢ (𝜑 → ((Homf ‘𝐶) = (Homf ‘𝐷) ↔ ∀𝑥 ∈ (Base‘𝐶)∀𝑦 ∈ (Base‘𝐶)(𝑥(Hom ‘𝐶)𝑦) = (𝑥(Hom ‘𝐷)𝑦))) |
10 | 4, 9 | mpbird 259 | 1 ⊢ (𝜑 → (Homf ‘𝐶) = (Homf ‘𝐷)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∀wral 3138 ‘cfv 6350 (class class class)co 7150 Basecbs 16477 Hom chom 16570 Homf chomf 16931 |
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 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 ax-rep 5183 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5322 ax-un 7455 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 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 3497 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 4562 df-pr 4564 df-op 4568 df-uni 4833 df-iun 4914 df-br 5060 df-opab 5122 df-mpt 5140 df-id 5455 df-xp 5556 df-rel 5557 df-cnv 5558 df-co 5559 df-dm 5560 df-rn 5561 df-res 5562 df-ima 5563 df-iota 6309 df-fun 6352 df-fn 6353 df-f 6354 df-f1 6355 df-fo 6356 df-f1o 6357 df-fv 6358 df-ov 7153 df-oprab 7154 df-mpo 7155 df-1st 7683 df-2nd 7684 df-homf 16935 |
This theorem is referenced by: (None) |
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