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Theorem gicrcl 19192

Description: Isomorphism implies the right side is a group. (Contributed by Mario Carneiro, 6-May-2015.)

**Assertion**
Ref	Expression
gicrcl	⊢ (𝑅 ≃_𝑔 𝑆 → 𝑆 ∈ Grp)

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

Step	Hyp	Ref	Expression
1		brgic 19188	. . 3 ⊢ (𝑅 ≃_𝑔 𝑆 ↔ (𝑅 GrpIso 𝑆) ≠ ∅)
2		n0 4302	. . 3 ⊢ ((𝑅 GrpIso 𝑆) ≠ ∅ ↔ ∃𝑓 𝑓 ∈ (𝑅 GrpIso 𝑆))
3	1, 2	bitri 275	. 2 ⊢ (𝑅 ≃_𝑔 𝑆 ↔ ∃𝑓 𝑓 ∈ (𝑅 GrpIso 𝑆))
4		gimghm 19182	. . . 4 ⊢ (𝑓 ∈ (𝑅 GrpIso 𝑆) → 𝑓 ∈ (𝑅 GrpHom 𝑆))
5		ghmgrp2 19137	. . . 4 ⊢ (𝑓 ∈ (𝑅 GrpHom 𝑆) → 𝑆 ∈ Grp)
6	4, 5	syl 17	. . 3 ⊢ (𝑓 ∈ (𝑅 GrpIso 𝑆) → 𝑆 ∈ Grp)
7	6	exlimiv 1931	. 2 ⊢ (∃𝑓 𝑓 ∈ (𝑅 GrpIso 𝑆) → 𝑆 ∈ Grp)
8	3, 7	sylbi 217	1 ⊢ (𝑅 ≃_𝑔 𝑆 → 𝑆 ∈ Grp)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∃wex 1780 ∈ wcel 2111 ≠ wne 2928 ∅c0 4282 class class class wbr 5093 (class class class)co 7352 Grpcgrp 18852 GrpHom cghm 19130 GrpIso cgim 19175 ≃_𝑔 cgic 19176

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 1968 ax-7 2009 ax-8 2113 ax-9 2121 ax-10 2144 ax-11 2160 ax-12 2180 ax-ext 2703 ax-sep 5236 ax-nul 5246 ax-pow 5305 ax-pr 5372 ax-un 7674

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2535 df-eu 2564 df-clab 2710 df-cleq 2723 df-clel 2806 df-nfc 2881 df-ne 2929 df-ral 3048 df-rex 3057 df-rab 3396 df-v 3438 df-sbc 3737 df-csb 3846 df-dif 3900 df-un 3902 df-in 3904 df-ss 3914 df-nul 4283 df-if 4475 df-pw 4551 df-sn 4576 df-pr 4578 df-op 4582 df-uni 4859 df-iun 4943 df-br 5094 df-opab 5156 df-mpt 5175 df-id 5514 df-xp 5625 df-rel 5626 df-cnv 5627 df-co 5628 df-dm 5629 df-rn 5630 df-res 5631 df-ima 5632 df-suc 6318 df-iota 6443 df-fun 6489 df-fn 6490 df-f 6491 df-f1 6492 df-fo 6493 df-f1o 6494 df-fv 6495 df-ov 7355 df-oprab 7356 df-mpo 7357 df-1st 7927 df-2nd 7928 df-1o 8391 df-map 8758 df-ghm 19131 df-gim 19177 df-gic 19178

This theorem is referenced by: (None)

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