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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 18409 | . 2 ⊢ (𝑅 ≃𝑔 𝑆 ↔ (𝑅 GrpIso 𝑆) ≠ ∅) | |
2 | n0 4310 | . . 3 ⊢ ((𝑅 GrpIso 𝑆) ≠ ∅ ↔ ∃𝑓 𝑓 ∈ (𝑅 GrpIso 𝑆)) | |
3 | gimcnv 18407 | . . . . 5 ⊢ (𝑓 ∈ (𝑅 GrpIso 𝑆) → ◡𝑓 ∈ (𝑆 GrpIso 𝑅)) | |
4 | brgici 18410 | . . . . 5 ⊢ (◡𝑓 ∈ (𝑆 GrpIso 𝑅) → 𝑆 ≃𝑔 𝑅) | |
5 | 3, 4 | syl 17 | . . . 4 ⊢ (𝑓 ∈ (𝑅 GrpIso 𝑆) → 𝑆 ≃𝑔 𝑅) |
6 | 5 | exlimiv 1931 | . . 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 1780 ∈ wcel 2114 ≠ wne 3016 ∅c0 4291 class class class wbr 5066 ◡ccnv 5554 (class class class)co 7156 GrpIso cgim 18397 ≃𝑔 cgic 18398 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 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 3496 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 4568 df-pr 4570 df-op 4574 df-uni 4839 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-id 5460 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-ov 7159 df-oprab 7160 df-mpo 7161 df-1st 7689 df-2nd 7690 df-1o 8102 df-mgm 17852 df-sgrp 17901 df-mnd 17912 df-grp 18106 df-ghm 18356 df-gim 18399 df-gic 18400 |
This theorem is referenced by: gicer 18416 cygznlem3 20716 cygth 20718 cyggic 20719 |
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