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Theorem gimcnv 19135
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 2732 . . . . . 6 (Base‘𝑆) = (Base‘𝑆)
2 eqid 2732 . . . . . 6 (Base‘𝑇) = (Base‘𝑇)
31, 2ghmf 19090 . . . . 5 (𝐹 ∈ (𝑆 GrpHom 𝑇) → 𝐹:(Base‘𝑆)⟶(Base‘𝑇))
4 frel 6719 . . . . . 6 (𝐹:(Base‘𝑆)⟶(Base‘𝑇) → Rel 𝐹)
5 dfrel2 6185 . . . . . 6 (Rel 𝐹𝐹 = 𝐹)
64, 5sylib 217 . . . . 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 2833 . . 3 (𝐹 ∈ (𝑆 GrpHom 𝑇) → 𝐹 ∈ (𝑆 GrpHom 𝑇))
109anim1ci 616 . 2 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐹 ∈ (𝑇 GrpHom 𝑆)) → (𝐹 ∈ (𝑇 GrpHom 𝑆) ∧ 𝐹 ∈ (𝑆 GrpHom 𝑇)))
11 isgim2 19133 . 2 (𝐹 ∈ (𝑆 GrpIso 𝑇) ↔ (𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐹 ∈ (𝑇 GrpHom 𝑆)))
12 isgim2 19133 . 2 (𝐹 ∈ (𝑇 GrpIso 𝑆) ↔ (𝐹 ∈ (𝑇 GrpHom 𝑆) ∧ 𝐹 ∈ (𝑆 GrpHom 𝑇)))
1310, 11, 123imtr4i 291 1 (𝐹 ∈ (𝑆 GrpIso 𝑇) → 𝐹 ∈ (𝑇 GrpIso 𝑆))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1541  wcel 2106  ccnv 5674  Rel wrel 5680  wf 6536  cfv 6540  (class class class)co 7405  Basecbs 17140   GrpHom cghm 19083   GrpIso cgim 19125
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7721
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-ov 7408  df-oprab 7409  df-mpo 7410  df-mgm 18557  df-sgrp 18606  df-mnd 18622  df-grp 18818  df-ghm 19084  df-gim 19127
This theorem is referenced by:  gicsym  19142  reloggim  26098  abliso  32184
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