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Mirrors > Home > MPE Home > Th. List > isohom | Structured version Visualization version GIF version |
Description: An isomorphism is a homomorphism. (Contributed by Mario Carneiro, 27-Jan-2017.) |
Ref | Expression |
---|---|
isohom.b | ⊢ 𝐵 = (Base‘𝐶) |
isohom.h | ⊢ 𝐻 = (Hom ‘𝐶) |
isohom.i | ⊢ 𝐼 = (Iso‘𝐶) |
isohom.c | ⊢ (𝜑 → 𝐶 ∈ Cat) |
isohom.x | ⊢ (𝜑 → 𝑋 ∈ 𝐵) |
isohom.y | ⊢ (𝜑 → 𝑌 ∈ 𝐵) |
Ref | Expression |
---|---|
isohom | ⊢ (𝜑 → (𝑋𝐼𝑌) ⊆ (𝑋𝐻𝑌)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | isohom.b | . . . 4 ⊢ 𝐵 = (Base‘𝐶) | |
2 | eqid 2778 | . . . 4 ⊢ (Inv‘𝐶) = (Inv‘𝐶) | |
3 | isohom.c | . . . 4 ⊢ (𝜑 → 𝐶 ∈ Cat) | |
4 | isohom.x | . . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝐵) | |
5 | isohom.y | . . . 4 ⊢ (𝜑 → 𝑌 ∈ 𝐵) | |
6 | isohom.i | . . . 4 ⊢ 𝐼 = (Iso‘𝐶) | |
7 | 1, 2, 3, 4, 5, 6 | isoval 16814 | . . 3 ⊢ (𝜑 → (𝑋𝐼𝑌) = dom (𝑋(Inv‘𝐶)𝑌)) |
8 | isohom.h | . . . . 5 ⊢ 𝐻 = (Hom ‘𝐶) | |
9 | 1, 2, 3, 4, 5, 8 | invss 16810 | . . . 4 ⊢ (𝜑 → (𝑋(Inv‘𝐶)𝑌) ⊆ ((𝑋𝐻𝑌) × (𝑌𝐻𝑋))) |
10 | dmss 5570 | . . . 4 ⊢ ((𝑋(Inv‘𝐶)𝑌) ⊆ ((𝑋𝐻𝑌) × (𝑌𝐻𝑋)) → dom (𝑋(Inv‘𝐶)𝑌) ⊆ dom ((𝑋𝐻𝑌) × (𝑌𝐻𝑋))) | |
11 | 9, 10 | syl 17 | . . 3 ⊢ (𝜑 → dom (𝑋(Inv‘𝐶)𝑌) ⊆ dom ((𝑋𝐻𝑌) × (𝑌𝐻𝑋))) |
12 | 7, 11 | eqsstrd 3858 | . 2 ⊢ (𝜑 → (𝑋𝐼𝑌) ⊆ dom ((𝑋𝐻𝑌) × (𝑌𝐻𝑋))) |
13 | dmxpss 5821 | . 2 ⊢ dom ((𝑋𝐻𝑌) × (𝑌𝐻𝑋)) ⊆ (𝑋𝐻𝑌) | |
14 | 12, 13 | syl6ss 3833 | 1 ⊢ (𝜑 → (𝑋𝐼𝑌) ⊆ (𝑋𝐻𝑌)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1601 ∈ wcel 2107 ⊆ wss 3792 × cxp 5355 dom cdm 5357 ‘cfv 6137 (class class class)co 6924 Basecbs 16259 Hom chom 16353 Catccat 16714 Invcinv 16794 Isociso 16795 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-8 2109 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-rep 5008 ax-sep 5019 ax-nul 5027 ax-pow 5079 ax-pr 5140 ax-un 7228 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ne 2970 df-ral 3095 df-rex 3096 df-reu 3097 df-rab 3099 df-v 3400 df-sbc 3653 df-csb 3752 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-nul 4142 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-op 4405 df-uni 4674 df-iun 4757 df-br 4889 df-opab 4951 df-mpt 4968 df-id 5263 df-xp 5363 df-rel 5364 df-cnv 5365 df-co 5366 df-dm 5367 df-rn 5368 df-res 5369 df-ima 5370 df-iota 6101 df-fun 6139 df-fn 6140 df-f 6141 df-f1 6142 df-fo 6143 df-f1o 6144 df-fv 6145 df-ov 6927 df-oprab 6928 df-mpt2 6929 df-1st 7447 df-2nd 7448 df-sect 16796 df-inv 16797 df-iso 16798 |
This theorem is referenced by: invisoinvl 16839 invcoisoid 16841 isocoinvid 16842 rcaninv 16843 ffthiso 16978 fuciso 17024 initoeu1 17050 initoeu2lem0 17052 initoeu2lem1 17053 initoeu2 17055 termoeu1 17057 nzerooringczr 43097 |
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