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Theorem gicerOLD 17647
Description: Obsolete proof of gicer 17646 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.
StepHypRef Expression
1 df-gic 17630 . . . . . 6 𝑔 = ( GrpIso “ (V ∖ 1𝑜))
2 cnvimass 5449 . . . . . . 7 ( GrpIso “ (V ∖ 1𝑜)) ⊆ dom GrpIso
3 gimfn 17631 . . . . . . . 8 GrpIso Fn (Grp × Grp)
4 fndm 5953 . . . . . . . 8 ( GrpIso Fn (Grp × Grp) → dom GrpIso = (Grp × Grp))
53, 4ax-mp 5 . . . . . . 7 dom GrpIso = (Grp × Grp)
62, 5sseqtri 3621 . . . . . 6 ( GrpIso “ (V ∖ 1𝑜)) ⊆ (Grp × Grp)
71, 6eqsstri 3619 . . . . 5 𝑔 ⊆ (Grp × Grp)
8 relxp 5193 . . . . 5 Rel (Grp × Grp)
9 relss 5172 . . . . 5 ( ≃𝑔 ⊆ (Grp × Grp) → (Rel (Grp × Grp) → Rel ≃𝑔 ))
107, 8, 9mp2 9 . . . 4 Rel ≃𝑔
1110a1i 11 . . 3 (⊤ → Rel ≃𝑔 )
12 gicsym 17644 . . . 4 (𝑥𝑔 𝑦𝑦𝑔 𝑥)
1312adantl 482 . . 3 ((⊤ ∧ 𝑥𝑔 𝑦) → 𝑦𝑔 𝑥)
14 gictr 17645 . . . 4 ((𝑥𝑔 𝑦𝑦𝑔 𝑧) → 𝑥𝑔 𝑧)
1514adantl 482 . . 3 ((⊤ ∧ (𝑥𝑔 𝑦𝑦𝑔 𝑧)) → 𝑥𝑔 𝑧)
16 gicref 17641 . . . . 5 (𝑥 ∈ Grp → 𝑥𝑔 𝑥)
17 giclcl 17642 . . . . 5 (𝑥𝑔 𝑥𝑥 ∈ Grp)
1816, 17impbii 199 . . . 4 (𝑥 ∈ Grp ↔ 𝑥𝑔 𝑥)
1918a1i 11 . . 3 (⊤ → (𝑥 ∈ Grp ↔ 𝑥𝑔 𝑥))
2011, 13, 15, 19iserd 7720 . 2 (⊤ → ≃𝑔 Er Grp)
2120trud 1490 1 𝑔 Er Grp
Colors of variables: wff setvar class
Syntax hints:  wb 196  wa 384   = wceq 1480  wtru 1481  wcel 1987  Vcvv 3189  cdif 3556  wss 3559   class class class wbr 4618   × cxp 5077  ccnv 5078  dom cdm 5079  cima 5082  Rel wrel 5084   Fn wfn 5847  1𝑜c1o 7505   Er wer 7691  Grpcgrp 17350   GrpIso cgim 17627  𝑔 cgic 17628
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4736  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6909
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rmo 2915  df-rab 2916  df-v 3191  df-sbc 3422  df-csb 3519  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-id 4994  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-suc 5693  df-iota 5815  df-fun 5854  df-fn 5855  df-f 5856  df-f1 5857  df-fo 5858  df-f1o 5859  df-fv 5860  df-riota 6571  df-ov 6613  df-oprab 6614  df-mpt2 6615  df-1st 7120  df-2nd 7121  df-1o 7512  df-er 7694  df-map 7811  df-0g 16030  df-mgm 17170  df-sgrp 17212  df-mnd 17223  df-mhm 17263  df-grp 17353  df-ghm 17586  df-gim 17629  df-gic 17630
This theorem is referenced by: (None)
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