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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 2728 | . . . 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 17755 | . . 3 ⊢ (𝜑 → (𝑋𝐼𝑌) = dom (𝑋(Inv‘𝐶)𝑌)) |
8 | isohom.h | . . . . 5 ⊢ 𝐻 = (Hom ‘𝐶) | |
9 | 1, 2, 3, 4, 5, 8 | invss 17751 | . . . 4 ⊢ (𝜑 → (𝑋(Inv‘𝐶)𝑌) ⊆ ((𝑋𝐻𝑌) × (𝑌𝐻𝑋))) |
10 | dmss 5909 | . . . 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 4020 | . 2 ⊢ (𝜑 → (𝑋𝐼𝑌) ⊆ dom ((𝑋𝐻𝑌) × (𝑌𝐻𝑋))) |
13 | dmxpss 6180 | . 2 ⊢ dom ((𝑋𝐻𝑌) × (𝑌𝐻𝑋)) ⊆ (𝑋𝐻𝑌) | |
14 | 12, 13 | sstrdi 3994 | 1 ⊢ (𝜑 → (𝑋𝐼𝑌) ⊆ (𝑋𝐻𝑌)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2098 ⊆ wss 3949 × cxp 5680 dom cdm 5682 ‘cfv 6553 (class class class)co 7426 Basecbs 17187 Hom chom 17251 Catccat 17651 Invcinv 17735 Isociso 17736 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2699 ax-rep 5289 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7746 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-ral 3059 df-rex 3068 df-reu 3375 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-iun 5002 df-br 5153 df-opab 5215 df-mpt 5236 df-id 5580 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-ov 7429 df-oprab 7430 df-mpo 7431 df-1st 7999 df-2nd 8000 df-sect 17737 df-inv 17738 df-iso 17739 |
This theorem is referenced by: invisoinvl 17780 invcoisoid 17782 isocoinvid 17783 rcaninv 17784 ffthiso 17925 fuciso 17974 initoeu1 18007 initoeu2lem0 18009 initoeu2lem1 18010 initoeu2 18012 termoeu1 18014 nzerooringczr 21413 thinccic 48145 |
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