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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 19051 | . . . 4 ⊢ ≃𝑔 = (◡ GrpIso “ (V ∖ 1o)) | |
2 | cnvimass 6034 | . . . . 5 ⊢ (◡ GrpIso “ (V ∖ 1o)) ⊆ dom GrpIso | |
3 | gimfn 19052 | . . . . . 6 ⊢ GrpIso Fn (Grp × Grp) | |
4 | 3 | fndmi 6607 | . . . . 5 ⊢ dom GrpIso = (Grp × Grp) |
5 | 2, 4 | sseqtri 3981 | . . . 4 ⊢ (◡ GrpIso “ (V ∖ 1o)) ⊆ (Grp × Grp) |
6 | 1, 5 | eqsstri 3979 | . . 3 ⊢ ≃𝑔 ⊆ (Grp × Grp) |
7 | relxp 5652 | . . 3 ⊢ Rel (Grp × Grp) | |
8 | relss 5738 | . . 3 ⊢ ( ≃𝑔 ⊆ (Grp × Grp) → (Rel (Grp × Grp) → Rel ≃𝑔 )) | |
9 | 6, 7, 8 | mp2 9 | . 2 ⊢ Rel ≃𝑔 |
10 | gicsym 19065 | . 2 ⊢ (𝑥 ≃𝑔 𝑦 → 𝑦 ≃𝑔 𝑥) | |
11 | gictr 19066 | . 2 ⊢ ((𝑥 ≃𝑔 𝑦 ∧ 𝑦 ≃𝑔 𝑧) → 𝑥 ≃𝑔 𝑧) | |
12 | gicref 19062 | . . 3 ⊢ (𝑥 ∈ Grp → 𝑥 ≃𝑔 𝑥) | |
13 | giclcl 19063 | . . 3 ⊢ (𝑥 ≃𝑔 𝑥 → 𝑥 ∈ Grp) | |
14 | 12, 13 | impbii 208 | . 2 ⊢ (𝑥 ∈ Grp ↔ 𝑥 ≃𝑔 𝑥) |
15 | 9, 10, 11, 14 | iseri 8676 | 1 ⊢ ≃𝑔 Er Grp |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2107 Vcvv 3446 ∖ cdif 3908 ⊆ wss 3911 class class class wbr 5106 × cxp 5632 ◡ccnv 5633 dom cdm 5634 “ cima 5637 Rel wrel 5639 1oc1o 8406 Er wer 8646 Grpcgrp 18749 GrpIso cgim 19048 ≃𝑔 cgic 19049 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2708 ax-rep 5243 ax-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-ral 3066 df-rex 3075 df-rmo 3354 df-reu 3355 df-rab 3409 df-v 3448 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-iun 4957 df-br 5107 df-opab 5169 df-mpt 5190 df-id 5532 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-suc 6324 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 df-riota 7314 df-ov 7361 df-oprab 7362 df-mpo 7363 df-1st 7922 df-2nd 7923 df-1o 8413 df-er 8649 df-map 8768 df-0g 17324 df-mgm 18498 df-sgrp 18547 df-mnd 18558 df-mhm 18602 df-grp 18752 df-ghm 19007 df-gim 19050 df-gic 19051 |
This theorem is referenced by: (None) |
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