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Mirrors > Home > MPE Home > Th. List > gicsym | Structured version Visualization version GIF version |
Description: Isomorphism is symmetric. (Contributed by Mario Carneiro, 21-Apr-2016.) |
Ref | Expression |
---|---|
gicsym | ⊢ (𝑅 ≃𝑔 𝑆 → 𝑆 ≃𝑔 𝑅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | brgic 18403 | . 2 ⊢ (𝑅 ≃𝑔 𝑆 ↔ (𝑅 GrpIso 𝑆) ≠ ∅) | |
2 | n0 4310 | . . 3 ⊢ ((𝑅 GrpIso 𝑆) ≠ ∅ ↔ ∃𝑓 𝑓 ∈ (𝑅 GrpIso 𝑆)) | |
3 | gimcnv 18401 | . . . . 5 ⊢ (𝑓 ∈ (𝑅 GrpIso 𝑆) → ◡𝑓 ∈ (𝑆 GrpIso 𝑅)) | |
4 | brgici 18404 | . . . . 5 ⊢ (◡𝑓 ∈ (𝑆 GrpIso 𝑅) → 𝑆 ≃𝑔 𝑅) | |
5 | 3, 4 | syl 17 | . . . 4 ⊢ (𝑓 ∈ (𝑅 GrpIso 𝑆) → 𝑆 ≃𝑔 𝑅) |
6 | 5 | exlimiv 1927 | . . 3 ⊢ (∃𝑓 𝑓 ∈ (𝑅 GrpIso 𝑆) → 𝑆 ≃𝑔 𝑅) |
7 | 2, 6 | sylbi 219 | . 2 ⊢ ((𝑅 GrpIso 𝑆) ≠ ∅ → 𝑆 ≃𝑔 𝑅) |
8 | 1, 7 | sylbi 219 | 1 ⊢ (𝑅 ≃𝑔 𝑆 → 𝑆 ≃𝑔 𝑅) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∃wex 1776 ∈ wcel 2110 ≠ wne 3016 ∅c0 4291 class class class wbr 5059 ◡ccnv 5549 (class class class)co 7150 GrpIso cgim 18391 ≃𝑔 cgic 18392 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 ax-rep 5183 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5322 ax-un 7455 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 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 3497 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 4562 df-pr 4564 df-op 4568 df-uni 4833 df-iun 4914 df-br 5060 df-opab 5122 df-mpt 5140 df-id 5455 df-xp 5556 df-rel 5557 df-cnv 5558 df-co 5559 df-dm 5560 df-rn 5561 df-res 5562 df-ima 5563 df-suc 6192 df-iota 6309 df-fun 6352 df-fn 6353 df-f 6354 df-f1 6355 df-fo 6356 df-f1o 6357 df-fv 6358 df-ov 7153 df-oprab 7154 df-mpo 7155 df-1st 7683 df-2nd 7684 df-1o 8096 df-mgm 17846 df-sgrp 17895 df-mnd 17906 df-grp 18100 df-ghm 18350 df-gim 18393 df-gic 18394 |
This theorem is referenced by: gicer 18410 cygznlem3 20710 cygth 20712 cyggic 20713 |
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