MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  gicer Structured version   Visualization version   GIF version

Theorem gicer 18892
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.
StepHypRef Expression
1 df-gic 18876 . . . 4 𝑔 = ( GrpIso “ (V ∖ 1o))
2 cnvimass 5989 . . . . 5 ( GrpIso “ (V ∖ 1o)) ⊆ dom GrpIso
3 gimfn 18877 . . . . . 6 GrpIso Fn (Grp × Grp)
43fndmi 6537 . . . . 5 dom GrpIso = (Grp × Grp)
52, 4sseqtri 3957 . . . 4 ( GrpIso “ (V ∖ 1o)) ⊆ (Grp × Grp)
61, 5eqsstri 3955 . . 3 𝑔 ⊆ (Grp × Grp)
7 relxp 5607 . . 3 Rel (Grp × Grp)
8 relss 5692 . . 3 ( ≃𝑔 ⊆ (Grp × Grp) → (Rel (Grp × Grp) → Rel ≃𝑔 ))
96, 7, 8mp2 9 . 2 Rel ≃𝑔
10 gicsym 18890 . 2 (𝑥𝑔 𝑦𝑦𝑔 𝑥)
11 gictr 18891 . 2 ((𝑥𝑔 𝑦𝑦𝑔 𝑧) → 𝑥𝑔 𝑧)
12 gicref 18887 . . 3 (𝑥 ∈ Grp → 𝑥𝑔 𝑥)
13 giclcl 18888 . . 3 (𝑥𝑔 𝑥𝑥 ∈ Grp)
1412, 13impbii 208 . 2 (𝑥 ∈ Grp ↔ 𝑥𝑔 𝑥)
159, 10, 11, 14iseri 8525 1 𝑔 Er Grp
Colors of variables: wff setvar class
Syntax hints:  wcel 2106  Vcvv 3432  cdif 3884  wss 3887   class class class wbr 5074   × cxp 5587  ccnv 5588  dom cdm 5589  cima 5592  Rel wrel 5594  1oc1o 8290   Er wer 8495  Grpcgrp 18577   GrpIso cgim 18873  𝑔 cgic 18874
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-rmo 3071  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-suc 6272  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-riota 7232  df-ov 7278  df-oprab 7279  df-mpo 7280  df-1st 7831  df-2nd 7832  df-1o 8297  df-er 8498  df-map 8617  df-0g 17152  df-mgm 18326  df-sgrp 18375  df-mnd 18386  df-mhm 18430  df-grp 18580  df-ghm 18832  df-gim 18875  df-gic 18876
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator