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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 2821 | . . . 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 17029 | . . 3 ⊢ (𝜑 → (𝑋𝐼𝑌) = dom (𝑋(Inv‘𝐶)𝑌)) |
8 | isohom.h | . . . . 5 ⊢ 𝐻 = (Hom ‘𝐶) | |
9 | 1, 2, 3, 4, 5, 8 | invss 17025 | . . . 4 ⊢ (𝜑 → (𝑋(Inv‘𝐶)𝑌) ⊆ ((𝑋𝐻𝑌) × (𝑌𝐻𝑋))) |
10 | dmss 5765 | . . . 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 4004 | . 2 ⊢ (𝜑 → (𝑋𝐼𝑌) ⊆ dom ((𝑋𝐻𝑌) × (𝑌𝐻𝑋))) |
13 | dmxpss 6022 | . 2 ⊢ dom ((𝑋𝐻𝑌) × (𝑌𝐻𝑋)) ⊆ (𝑋𝐻𝑌) | |
14 | 12, 13 | sstrdi 3978 | 1 ⊢ (𝜑 → (𝑋𝐼𝑌) ⊆ (𝑋𝐻𝑌)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2110 ⊆ wss 3935 × cxp 5547 dom cdm 5549 ‘cfv 6349 (class class class)co 7150 Basecbs 16477 Hom chom 16570 Catccat 16929 Invcinv 17009 Isociso 17010 |
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 2157 ax-12 2173 ax-ext 2793 ax-rep 5182 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 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 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-op 4567 df-uni 4832 df-iun 4913 df-br 5059 df-opab 5121 df-mpt 5139 df-id 5454 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-ov 7153 df-oprab 7154 df-mpo 7155 df-1st 7683 df-2nd 7684 df-sect 17011 df-inv 17012 df-iso 17013 |
This theorem is referenced by: invisoinvl 17054 invcoisoid 17056 isocoinvid 17057 rcaninv 17058 ffthiso 17193 fuciso 17239 initoeu1 17265 initoeu2lem0 17267 initoeu2lem1 17268 initoeu2 17270 termoeu1 17272 nzerooringczr 44337 |
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