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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
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