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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 17035 | . . 3 ⊢ (𝜑 → (𝑋𝐼𝑌) = dom (𝑋(Inv‘𝐶)𝑌)) |
8 | isohom.h | . . . . 5 ⊢ 𝐻 = (Hom ‘𝐶) | |
9 | 1, 2, 3, 4, 5, 8 | invss 17031 | . . . 4 ⊢ (𝜑 → (𝑋(Inv‘𝐶)𝑌) ⊆ ((𝑋𝐻𝑌) × (𝑌𝐻𝑋))) |
10 | dmss 5771 | . . . 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 4005 | . 2 ⊢ (𝜑 → (𝑋𝐼𝑌) ⊆ dom ((𝑋𝐻𝑌) × (𝑌𝐻𝑋))) |
13 | dmxpss 6028 | . 2 ⊢ dom ((𝑋𝐻𝑌) × (𝑌𝐻𝑋)) ⊆ (𝑋𝐻𝑌) | |
14 | 12, 13 | sstrdi 3979 | 1 ⊢ (𝜑 → (𝑋𝐼𝑌) ⊆ (𝑋𝐻𝑌)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2114 ⊆ wss 3936 × cxp 5553 dom cdm 5555 ‘cfv 6355 (class class class)co 7156 Basecbs 16483 Hom chom 16576 Catccat 16935 Invcinv 17015 Isociso 17016 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 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 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 4568 df-pr 4570 df-op 4574 df-uni 4839 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-id 5460 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-ov 7159 df-oprab 7160 df-mpo 7161 df-1st 7689 df-2nd 7690 df-sect 17017 df-inv 17018 df-iso 17019 |
This theorem is referenced by: invisoinvl 17060 invcoisoid 17062 isocoinvid 17063 rcaninv 17064 ffthiso 17199 fuciso 17245 initoeu1 17271 initoeu2lem0 17273 initoeu2lem1 17274 initoeu2 17276 termoeu1 17278 nzerooringczr 44363 |
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