Theorem gicerOLD 17719

Description: Obsolete proof of gicer 17718 as of 1-May-2021. Isomorphism is an equivalence relation on groups. (Contributed by Mario Carneiro, 21-Apr-2016.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
gicerOLD	⊢ ≃𝑔 Er Grp

Proof of Theorem gicerOLD

Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-gic 17702	. . . . . 6 ⊢ ≃𝑔 = (◡ GrpIso “ (V ∖ 1𝑜))
2		cnvimass 5485	. . . . . . 7 ⊢ (◡ GrpIso “ (V ∖ 1𝑜)) ⊆ dom GrpIso
3		gimfn 17703	. . . . . . . 8 ⊢ GrpIso Fn (Grp × Grp)
4		fndm 5990	. . . . . . . 8 ⊢ ( GrpIso Fn (Grp × Grp) → dom GrpIso = (Grp × Grp))
5	3, 4	ax-mp 5	. . . . . . 7 ⊢ dom GrpIso = (Grp × Grp)
6	2, 5	sseqtri 3637	. . . . . 6 ⊢ (◡ GrpIso “ (V ∖ 1𝑜)) ⊆ (Grp × Grp)
7	1, 6	eqsstri 3635	. . . . 5 ⊢ ≃𝑔 ⊆ (Grp × Grp)
8		relxp 5227	. . . . 5 ⊢ Rel (Grp × Grp)
9		relss 5206	. . . . 5 ⊢ ( ≃𝑔 ⊆ (Grp × Grp) → (Rel (Grp × Grp) → Rel ≃𝑔 ))
10	7, 8, 9	mp2 9	. . . 4 ⊢ Rel ≃𝑔
11	10	a1i 11	. . 3 ⊢ (⊤ → Rel ≃𝑔 )
12		gicsym 17716	. . . 4 ⊢ (𝑥 ≃𝑔 𝑦 → 𝑦 ≃𝑔 𝑥)
13	12	adantl 482	. . 3 ⊢ ((⊤ ∧ 𝑥 ≃𝑔 𝑦) → 𝑦 ≃𝑔 𝑥)
14		gictr 17717	. . . 4 ⊢ ((𝑥 ≃𝑔 𝑦 ∧ 𝑦 ≃𝑔 𝑧) → 𝑥 ≃𝑔 𝑧)
15	14	adantl 482	. . 3 ⊢ ((⊤ ∧ (𝑥 ≃𝑔 𝑦 ∧ 𝑦 ≃𝑔 𝑧)) → 𝑥 ≃𝑔 𝑧)
16		gicref 17713	. . . . 5 ⊢ (𝑥 ∈ Grp → 𝑥 ≃𝑔 𝑥)
17		giclcl 17714	. . . . 5 ⊢ (𝑥 ≃𝑔 𝑥 → 𝑥 ∈ Grp)
18	16, 17	impbii 199	. . . 4 ⊢ (𝑥 ∈ Grp ↔ 𝑥 ≃𝑔 𝑥)
19	18	a1i 11	. . 3 ⊢ (⊤ → (𝑥 ∈ Grp ↔ 𝑥 ≃𝑔 𝑥))
20	11, 13, 15, 19	iserd 7768	. 2 ⊢ (⊤ → ≃𝑔 Er Grp)
21	20	trud 1493	1 ⊢ ≃𝑔 Er Grp

Syntax hints:   ↔ wb 196   ∧ wa 384   = wceq 1483  ⊤wtru 1484   ∈ wcel 1990  Vcvv 3200   ∖ cdif 3571   ⊆ wss 3574   class class class wbr 4653   × cxp 5112  ^◡ccnv 5113  dom cdm 5114   “ cima 5117  Rel wrel 5119   Fn wfn 5883  1_𝑜c1o 7553   Er wer 7739  Grpcgrp 17422   GrpIso cgim 17699   ≃_𝑔 cgic 17700

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-1st 7168  df-2nd 7169  df-1o 7560  df-er 7742  df-map 7859  df-0g 16102  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-mhm 17335  df-grp 17425  df-ghm 17658  df-gim 17701  df-gic 17702

This theorem is referenced by: (None)