Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  isgim2 Structured version   Visualization version   GIF version

Theorem isgim2 17754
 Description: A group isomorphism is a homomorphism whose converse is also a homomorphism. Characterization of isomorphisms similar to ishmeo 21610. (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 2651 . . 3 (Base‘𝑅) = (Base‘𝑅)
2 eqid 2651 . . 3 (Base‘𝑆) = (Base‘𝑆)
31, 2isgim 17751 . 2 (𝐹 ∈ (𝑅 GrpIso 𝑆) ↔ (𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ 𝐹:(Base‘𝑅)–1-1-onto→(Base‘𝑆)))
41, 2ghmf1o 17737 . . 3 (𝐹 ∈ (𝑅 GrpHom 𝑆) → (𝐹:(Base‘𝑅)–1-1-onto→(Base‘𝑆) ↔ 𝐹 ∈ (𝑆 GrpHom 𝑅)))
54pm5.32i 670 . 2 ((𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ 𝐹:(Base‘𝑅)–1-1-onto→(Base‘𝑆)) ↔ (𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ 𝐹 ∈ (𝑆 GrpHom 𝑅)))
63, 5bitri 264 1 (𝐹 ∈ (𝑅 GrpIso 𝑆) ↔ (𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ 𝐹 ∈ (𝑆 GrpHom 𝑅)))
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 196   ∧ wa 383   ∈ wcel 2030  ◡ccnv 5142  –1-1-onto→wf1o 5925  ‘cfv 5926  (class class class)co 6690  Basecbs 15904   GrpHom cghm 17704   GrpIso cgim 17746 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-mgm 17289  df-sgrp 17331  df-mnd 17342  df-grp 17472  df-ghm 17705  df-gim 17748 This theorem is referenced by:  gimcnv  17756  gimco  17757  gicref  17760  pi1xfrgim  22904
 Copyright terms: Public domain W3C validator