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Theorem gicer 18418

Description: Isomorphism is an equivalence relation on groups. (Contributed by Mario Carneiro, 21-Apr-2016.) (Proof shortened by AV, 1-May-2021.)

**Assertion**
Ref	Expression
gicer	⊢ ≃_𝑔 Er Grp

Proof of Theorem gicer Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-gic 18402	. . . 4 ⊢ ≃_𝑔 = (^◡ GrpIso “ (V ∖ 1_o))
2		cnvimass 5951	. . . . 5 ⊢ (^◡ GrpIso “ (V ∖ 1_o)) ⊆ dom GrpIso
3		gimfn 18403	. . . . . 6 ⊢ GrpIso Fn (Grp × Grp)
4		fndm 6457	. . . . . 6 ⊢ ( GrpIso Fn (Grp × Grp) → dom GrpIso = (Grp × Grp))
5	3, 4	ax-mp 5	. . . . 5 ⊢ dom GrpIso = (Grp × Grp)
6	2, 5	sseqtri 4005	. . . 4 ⊢ (^◡ GrpIso “ (V ∖ 1_o)) ⊆ (Grp × Grp)
7	1, 6	eqsstri 4003	. . 3 ⊢ ≃_𝑔 ⊆ (Grp × Grp)
8		relxp 5575	. . 3 ⊢ Rel (Grp × Grp)
9		relss 5658	. . 3 ⊢ ( ≃_𝑔 ⊆ (Grp × Grp) → (Rel (Grp × Grp) → Rel ≃_𝑔 ))
10	7, 8, 9	mp2 9	. 2 ⊢ Rel ≃_𝑔
11		gicsym 18416	. 2 ⊢ (𝑥 ≃_𝑔 𝑦 → 𝑦 ≃_𝑔 𝑥)
12		gictr 18417	. 2 ⊢ ((𝑥 ≃_𝑔 𝑦 ∧ 𝑦 ≃_𝑔 𝑧) → 𝑥 ≃_𝑔 𝑧)
13		gicref 18413	. . 3 ⊢ (𝑥 ∈ Grp → 𝑥 ≃_𝑔 𝑥)
14		giclcl 18414	. . 3 ⊢ (𝑥 ≃_𝑔 𝑥 → 𝑥 ∈ Grp)
15	13, 14	impbii 211	. 2 ⊢ (𝑥 ∈ Grp ↔ 𝑥 ≃_𝑔 𝑥)
16	10, 11, 12, 15	iseri 8318	1 ⊢ ≃_𝑔 Er Grp

Colors of variables: wff setvar class

Syntax hints: = wceq 1537 ∈ wcel 2114 Vcvv 3496 ∖ cdif 3935 ⊆ wss 3938 class class class wbr 5068 × cxp 5555 ^◡ccnv 5556 dom cdm 5557 “ cima 5560 Rel wrel 5562 Fn wfn 6352 1_oc1o 8097 Er wer 8288 Grpcgrp 18105 GrpIso cgim 18399 ≃_𝑔 cgic 18400

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 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463

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 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-id 5462 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-1st 7691 df-2nd 7692 df-1o 8104 df-er 8291 df-map 8410 df-0g 16717 df-mgm 17854 df-sgrp 17903 df-mnd 17914 df-mhm 17958 df-grp 18108 df-ghm 18358 df-gim 18401 df-gic 18402

This theorem is referenced by: (None)

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