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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 18511 | . . . 4 ⊢ ≃𝑔 = (◡ GrpIso “ (V ∖ 1o)) | |
2 | cnvimass 5917 | . . . . 5 ⊢ (◡ GrpIso “ (V ∖ 1o)) ⊆ dom GrpIso | |
3 | gimfn 18512 | . . . . . 6 ⊢ GrpIso Fn (Grp × Grp) | |
4 | 3 | fndmi 6435 | . . . . 5 ⊢ dom GrpIso = (Grp × Grp) |
5 | 2, 4 | sseqtri 3911 | . . . 4 ⊢ (◡ GrpIso “ (V ∖ 1o)) ⊆ (Grp × Grp) |
6 | 1, 5 | eqsstri 3909 | . . 3 ⊢ ≃𝑔 ⊆ (Grp × Grp) |
7 | relxp 5537 | . . 3 ⊢ Rel (Grp × Grp) | |
8 | relss 5621 | . . 3 ⊢ ( ≃𝑔 ⊆ (Grp × Grp) → (Rel (Grp × Grp) → Rel ≃𝑔 )) | |
9 | 6, 7, 8 | mp2 9 | . 2 ⊢ Rel ≃𝑔 |
10 | gicsym 18525 | . 2 ⊢ (𝑥 ≃𝑔 𝑦 → 𝑦 ≃𝑔 𝑥) | |
11 | gictr 18526 | . 2 ⊢ ((𝑥 ≃𝑔 𝑦 ∧ 𝑦 ≃𝑔 𝑧) → 𝑥 ≃𝑔 𝑧) | |
12 | gicref 18522 | . . 3 ⊢ (𝑥 ∈ Grp → 𝑥 ≃𝑔 𝑥) | |
13 | giclcl 18523 | . . 3 ⊢ (𝑥 ≃𝑔 𝑥 → 𝑥 ∈ Grp) | |
14 | 12, 13 | impbii 212 | . 2 ⊢ (𝑥 ∈ Grp ↔ 𝑥 ≃𝑔 𝑥) |
15 | 9, 10, 11, 14 | iseri 8340 | 1 ⊢ ≃𝑔 Er Grp |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2113 Vcvv 3397 ∖ cdif 3838 ⊆ wss 3841 class class class wbr 5027 × cxp 5517 ◡ccnv 5518 dom cdm 5519 “ cima 5522 Rel wrel 5524 1oc1o 8117 Er wer 8310 Grpcgrp 18212 GrpIso cgim 18508 ≃𝑔 cgic 18509 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1916 ax-6 1974 ax-7 2019 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2161 ax-12 2178 ax-ext 2710 ax-rep 5151 ax-sep 5164 ax-nul 5171 ax-pow 5229 ax-pr 5293 ax-un 7473 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2074 df-mo 2540 df-eu 2570 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-ne 2935 df-ral 3058 df-rex 3059 df-reu 3060 df-rmo 3061 df-rab 3062 df-v 3399 df-sbc 3680 df-csb 3789 df-dif 3844 df-un 3846 df-in 3848 df-ss 3858 df-nul 4210 df-if 4412 df-pw 4487 df-sn 4514 df-pr 4516 df-op 4520 df-uni 4794 df-iun 4880 df-br 5028 df-opab 5090 df-mpt 5108 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-suc 6172 df-iota 6291 df-fun 6335 df-fn 6336 df-f 6337 df-f1 6338 df-fo 6339 df-f1o 6340 df-fv 6341 df-riota 7121 df-ov 7167 df-oprab 7168 df-mpo 7169 df-1st 7707 df-2nd 7708 df-1o 8124 df-er 8313 df-map 8432 df-0g 16811 df-mgm 17961 df-sgrp 18010 df-mnd 18021 df-mhm 18065 df-grp 18215 df-ghm 18467 df-gim 18510 df-gic 18511 |
This theorem is referenced by: (None) |
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