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Mirrors > Home > MPE Home > Th. List > gicer | Structured version Visualization version GIF version |
Description: Isomorphism is an equivalence relation on groups. (Contributed by Mario Carneiro, 21-Apr-2016.) (Proof shortened by AV, 1-May-2021.) |
Ref | Expression |
---|---|
gicer | ⊢ ≃𝑔 Er Grp |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-gic 19176 | . . . 4 ⊢ ≃𝑔 = (◡ GrpIso “ (V ∖ 1o)) | |
2 | cnvimass 6081 | . . . . 5 ⊢ (◡ GrpIso “ (V ∖ 1o)) ⊆ dom GrpIso | |
3 | gimfn 19177 | . . . . . 6 ⊢ GrpIso Fn (Grp × Grp) | |
4 | 3 | fndmi 6654 | . . . . 5 ⊢ dom GrpIso = (Grp × Grp) |
5 | 2, 4 | sseqtri 4019 | . . . 4 ⊢ (◡ GrpIso “ (V ∖ 1o)) ⊆ (Grp × Grp) |
6 | 1, 5 | eqsstri 4017 | . . 3 ⊢ ≃𝑔 ⊆ (Grp × Grp) |
7 | relxp 5695 | . . 3 ⊢ Rel (Grp × Grp) | |
8 | relss 5782 | . . 3 ⊢ ( ≃𝑔 ⊆ (Grp × Grp) → (Rel (Grp × Grp) → Rel ≃𝑔 )) | |
9 | 6, 7, 8 | mp2 9 | . 2 ⊢ Rel ≃𝑔 |
10 | gicsym 19191 | . 2 ⊢ (𝑥 ≃𝑔 𝑦 → 𝑦 ≃𝑔 𝑥) | |
11 | gictr 19192 | . 2 ⊢ ((𝑥 ≃𝑔 𝑦 ∧ 𝑦 ≃𝑔 𝑧) → 𝑥 ≃𝑔 𝑧) | |
12 | gicref 19188 | . . 3 ⊢ (𝑥 ∈ Grp → 𝑥 ≃𝑔 𝑥) | |
13 | giclcl 19189 | . . 3 ⊢ (𝑥 ≃𝑔 𝑥 → 𝑥 ∈ Grp) | |
14 | 12, 13 | impbii 208 | . 2 ⊢ (𝑥 ∈ Grp ↔ 𝑥 ≃𝑔 𝑥) |
15 | 9, 10, 11, 14 | iseri 8734 | 1 ⊢ ≃𝑔 Er Grp |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2104 Vcvv 3472 ∖ cdif 3946 ⊆ wss 3949 class class class wbr 5149 × cxp 5675 ◡ccnv 5676 dom cdm 5677 “ cima 5680 Rel wrel 5682 1oc1o 8463 Er wer 8704 Grpcgrp 18857 GrpIso cgim 19173 ≃𝑔 cgic 19174 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-10 2135 ax-11 2152 ax-12 2169 ax-ext 2701 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7729 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 844 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2532 df-eu 2561 df-clab 2708 df-cleq 2722 df-clel 2808 df-nfc 2883 df-ne 2939 df-ral 3060 df-rex 3069 df-rmo 3374 df-reu 3375 df-rab 3431 df-v 3474 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-iun 5000 df-br 5150 df-opab 5212 df-mpt 5233 df-id 5575 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-suc 6371 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-riota 7369 df-ov 7416 df-oprab 7417 df-mpo 7418 df-1st 7979 df-2nd 7980 df-1o 8470 df-er 8707 df-map 8826 df-0g 17393 df-mgm 18567 df-sgrp 18646 df-mnd 18662 df-mhm 18707 df-grp 18860 df-ghm 19130 df-gim 19175 df-gic 19176 |
This theorem is referenced by: (None) |
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