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

Theorem gimcnv 18398
Description: The converse of a bijective group homomorphism is a bijective group homomorphism. (Contributed by Stefan O'Rear, 25-Jan-2015.) (Revised by Mario Carneiro, 6-May-2015.)
Assertion
Ref Expression
gimcnv (𝐹 ∈ (𝑆 GrpIso 𝑇) → 𝐹 ∈ (𝑇 GrpIso 𝑆))

Proof of Theorem gimcnv
StepHypRef Expression
1 eqid 2822 . . . . . 6 (Base‘𝑆) = (Base‘𝑆)
2 eqid 2822 . . . . . 6 (Base‘𝑇) = (Base‘𝑇)
31, 2ghmf 18353 . . . . 5 (𝐹 ∈ (𝑆 GrpHom 𝑇) → 𝐹:(Base‘𝑆)⟶(Base‘𝑇))
4 frel 6499 . . . . . 6 (𝐹:(Base‘𝑆)⟶(Base‘𝑇) → Rel 𝐹)
5 dfrel2 6024 . . . . . 6 (Rel 𝐹𝐹 = 𝐹)
64, 5sylib 221 . . . . 5 (𝐹:(Base‘𝑆)⟶(Base‘𝑇) → 𝐹 = 𝐹)
73, 6syl 17 . . . 4 (𝐹 ∈ (𝑆 GrpHom 𝑇) → 𝐹 = 𝐹)
8 id 22 . . . 4 (𝐹 ∈ (𝑆 GrpHom 𝑇) → 𝐹 ∈ (𝑆 GrpHom 𝑇))
97, 8eqeltrd 2914 . . 3 (𝐹 ∈ (𝑆 GrpHom 𝑇) → 𝐹 ∈ (𝑆 GrpHom 𝑇))
109anim1ci 618 . 2 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐹 ∈ (𝑇 GrpHom 𝑆)) → (𝐹 ∈ (𝑇 GrpHom 𝑆) ∧ 𝐹 ∈ (𝑆 GrpHom 𝑇)))
11 isgim2 18396 . 2 (𝐹 ∈ (𝑆 GrpIso 𝑇) ↔ (𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐹 ∈ (𝑇 GrpHom 𝑆)))
12 isgim2 18396 . 2 (𝐹 ∈ (𝑇 GrpIso 𝑆) ↔ (𝐹 ∈ (𝑇 GrpHom 𝑆) ∧ 𝐹 ∈ (𝑆 GrpHom 𝑇)))
1310, 11, 123imtr4i 295 1 (𝐹 ∈ (𝑆 GrpIso 𝑇) → 𝐹 ∈ (𝑇 GrpIso 𝑆))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399   = wceq 1538  wcel 2114  ccnv 5531  Rel wrel 5537  wf 6330  cfv 6334  (class class class)co 7140  Basecbs 16474   GrpHom cghm 18346   GrpIso cgim 18388
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794  ax-rep 5166  ax-sep 5179  ax-nul 5186  ax-pow 5243  ax-pr 5307  ax-un 7446
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-ne 3012  df-ral 3135  df-rex 3136  df-reu 3137  df-rab 3139  df-v 3471  df-sbc 3748  df-csb 3856  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4266  df-if 4440  df-pw 4513  df-sn 4540  df-pr 4542  df-op 4546  df-uni 4814  df-iun 4896  df-br 5043  df-opab 5105  df-mpt 5123  df-id 5437  df-xp 5538  df-rel 5539  df-cnv 5540  df-co 5541  df-dm 5542  df-rn 5543  df-res 5544  df-ima 5545  df-iota 6293  df-fun 6336  df-fn 6337  df-f 6338  df-f1 6339  df-fo 6340  df-f1o 6341  df-fv 6342  df-ov 7143  df-oprab 7144  df-mpo 7145  df-mgm 17843  df-sgrp 17892  df-mnd 17903  df-grp 18097  df-ghm 18347  df-gim 18390
This theorem is referenced by:  gicsym  18405  reloggim  25188  abliso  30714
  Copyright terms: Public domain W3C validator