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Theorem isgim2 19056
Description: A group isomorphism is a homomorphism whose converse is also a homomorphism. Characterization of isomorphisms similar to ishmeo 23113. (Contributed by Mario Carneiro, 6-May-2015.)
Assertion
Ref Expression
isgim2 (𝐹 ∈ (𝑅 GrpIso 𝑆) ↔ (𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ 𝐹 ∈ (𝑆 GrpHom 𝑅)))

Proof of Theorem isgim2
StepHypRef Expression
1 eqid 2737 . . 3 (Base‘𝑅) = (Base‘𝑅)
2 eqid 2737 . . 3 (Base‘𝑆) = (Base‘𝑆)
31, 2isgim 19053 . 2 (𝐹 ∈ (𝑅 GrpIso 𝑆) ↔ (𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ 𝐹:(Base‘𝑅)–1-1-onto→(Base‘𝑆)))
41, 2ghmf1o 19039 . . 3 (𝐹 ∈ (𝑅 GrpHom 𝑆) → (𝐹:(Base‘𝑅)–1-1-onto→(Base‘𝑆) ↔ 𝐹 ∈ (𝑆 GrpHom 𝑅)))
54pm5.32i 576 . 2 ((𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ 𝐹:(Base‘𝑅)–1-1-onto→(Base‘𝑆)) ↔ (𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ 𝐹 ∈ (𝑆 GrpHom 𝑅)))
63, 5bitri 275 1 (𝐹 ∈ (𝑅 GrpIso 𝑆) ↔ (𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ 𝐹 ∈ (𝑆 GrpHom 𝑅)))
Colors of variables: wff setvar class
Syntax hints:  wb 205  wa 397  wcel 2107  ccnv 5633  1-1-ontowf1o 6496  cfv 6497  (class class class)co 7358  Basecbs 17084   GrpHom cghm 19006   GrpIso cgim 19048
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 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-rep 5243  ax-sep 5257  ax-nul 5264  ax-pow 5321  ax-pr 5385  ax-un 7673
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-ral 3066  df-rex 3075  df-reu 3355  df-rab 3409  df-v 3448  df-sbc 3741  df-csb 3857  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-iun 4957  df-br 5107  df-opab 5169  df-mpt 5190  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-f1 6502  df-fo 6503  df-f1o 6504  df-fv 6505  df-ov 7361  df-oprab 7362  df-mpo 7363  df-mgm 18498  df-sgrp 18547  df-mnd 18558  df-grp 18752  df-ghm 19007  df-gim 19050
This theorem is referenced by:  gimcnv  19058  gimco  19059  gicref  19062  pi1xfrgim  24424
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