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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 2798 | . . . 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 17027 | . . 3 ⊢ (𝜑 → (𝑋𝐼𝑌) = dom (𝑋(Inv‘𝐶)𝑌)) |
8 | isohom.h | . . . . 5 ⊢ 𝐻 = (Hom ‘𝐶) | |
9 | 1, 2, 3, 4, 5, 8 | invss 17023 | . . . 4 ⊢ (𝜑 → (𝑋(Inv‘𝐶)𝑌) ⊆ ((𝑋𝐻𝑌) × (𝑌𝐻𝑋))) |
10 | dmss 5735 | . . . 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 3953 | . 2 ⊢ (𝜑 → (𝑋𝐼𝑌) ⊆ dom ((𝑋𝐻𝑌) × (𝑌𝐻𝑋))) |
13 | dmxpss 5995 | . 2 ⊢ dom ((𝑋𝐻𝑌) × (𝑌𝐻𝑋)) ⊆ (𝑋𝐻𝑌) | |
14 | 12, 13 | sstrdi 3927 | 1 ⊢ (𝜑 → (𝑋𝐼𝑌) ⊆ (𝑋𝐻𝑌)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1538 ∈ wcel 2111 ⊆ wss 3881 × cxp 5517 dom cdm 5519 ‘cfv 6324 (class class class)co 7135 Basecbs 16475 Hom chom 16568 Catccat 16927 Invcinv 17007 Isociso 17008 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-rex 3112 df-reu 3113 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-ov 7138 df-oprab 7139 df-mpo 7140 df-1st 7671 df-2nd 7672 df-sect 17009 df-inv 17010 df-iso 17011 |
This theorem is referenced by: invisoinvl 17052 invcoisoid 17054 isocoinvid 17055 rcaninv 17056 ffthiso 17191 fuciso 17237 initoeu1 17263 initoeu2lem0 17265 initoeu2lem1 17266 initoeu2 17268 termoeu1 17270 nzerooringczr 44696 |
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