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Theorem gicsym 19220

Description: Isomorphism is symmetric. (Contributed by Mario Carneiro, 21-Apr-2016.)

**Assertion**
Ref	Expression
gicsym	⊢ (𝑅 ≃_𝑔 𝑆 → 𝑆 ≃_𝑔 𝑅)

Proof of Theorem gicsym Dummy variable 𝑓 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		brgic 19215	. 2 ⊢ (𝑅 ≃_𝑔 𝑆 ↔ (𝑅 GrpIso 𝑆) ≠ ∅)
2		n0 4342	. . 3 ⊢ ((𝑅 GrpIso 𝑆) ≠ ∅ ↔ ∃𝑓 𝑓 ∈ (𝑅 GrpIso 𝑆))
3		gimcnv 19212	. . . . 5 ⊢ (𝑓 ∈ (𝑅 GrpIso 𝑆) → ^◡𝑓 ∈ (𝑆 GrpIso 𝑅))
4		brgici 19216	. . . . 5 ⊢ (^◡𝑓 ∈ (𝑆 GrpIso 𝑅) → 𝑆 ≃_𝑔 𝑅)
5	3, 4	syl 17	. . . 4 ⊢ (𝑓 ∈ (𝑅 GrpIso 𝑆) → 𝑆 ≃_𝑔 𝑅)
6	5	exlimiv 1926	. . 3 ⊢ (∃𝑓 𝑓 ∈ (𝑅 GrpIso 𝑆) → 𝑆 ≃_𝑔 𝑅)
7	2, 6	sylbi 216	. 2 ⊢ ((𝑅 GrpIso 𝑆) ≠ ∅ → 𝑆 ≃_𝑔 𝑅)
8	1, 7	sylbi 216	1 ⊢ (𝑅 ≃_𝑔 𝑆 → 𝑆 ≃_𝑔 𝑅)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∃wex 1774 ∈ wcel 2099 ≠ wne 2935 ∅c0 4318 class class class wbr 5142 ^◡ccnv 5671 (class class class)co 7414 GrpIso cgim 19202 ≃_𝑔 cgic 19203

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2164 ax-ext 2698 ax-rep 5279 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7734

This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2936 df-ral 3057 df-rex 3066 df-reu 3372 df-rab 3428 df-v 3471 df-sbc 3775 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-iun 4993 df-br 5143 df-opab 5205 df-mpt 5226 df-id 5570 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-ov 7417 df-oprab 7418 df-mpo 7419 df-1st 7987 df-2nd 7988 df-1o 8480 df-mgm 18591 df-sgrp 18670 df-mnd 18686 df-grp 18884 df-ghm 19159 df-gim 19204 df-gic 19205

This theorem is referenced by: gicer 19222 cygznlem3 21490 cygth 21492 cyggic 21493

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