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Theorem isgim 17901
Description: An isomorphism of groups is a bijective homomorphism. (Contributed by Stefan O'Rear, 21-Jan-2015.)
Hypotheses
Ref Expression
isgim.b 𝐵 = (Base‘𝑅)
isgim.c 𝐶 = (Base‘𝑆)
Assertion
Ref Expression
isgim (𝐹 ∈ (𝑅 GrpIso 𝑆) ↔ (𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ 𝐹:𝐵1-1-onto𝐶))

Proof of Theorem isgim
Dummy variables 𝑎 𝑏 𝑐 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-3an 1074 . 2 ((𝑅 ∈ Grp ∧ 𝑆 ∈ Grp ∧ 𝐹 ∈ {𝑐 ∈ (𝑅 GrpHom 𝑆) ∣ 𝑐:𝐵1-1-onto𝐶}) ↔ ((𝑅 ∈ Grp ∧ 𝑆 ∈ Grp) ∧ 𝐹 ∈ {𝑐 ∈ (𝑅 GrpHom 𝑆) ∣ 𝑐:𝐵1-1-onto𝐶}))
2 df-gim 17898 . . 3 GrpIso = (𝑎 ∈ Grp, 𝑏 ∈ Grp ↦ {𝑐 ∈ (𝑎 GrpHom 𝑏) ∣ 𝑐:(Base‘𝑎)–1-1-onto→(Base‘𝑏)})
3 ovex 6837 . . . 4 (𝑎 GrpHom 𝑏) ∈ V
43rabex 4960 . . 3 {𝑐 ∈ (𝑎 GrpHom 𝑏) ∣ 𝑐:(Base‘𝑎)–1-1-onto→(Base‘𝑏)} ∈ V
5 oveq12 6818 . . . 4 ((𝑎 = 𝑅𝑏 = 𝑆) → (𝑎 GrpHom 𝑏) = (𝑅 GrpHom 𝑆))
6 fveq2 6348 . . . . . 6 (𝑎 = 𝑅 → (Base‘𝑎) = (Base‘𝑅))
7 isgim.b . . . . . 6 𝐵 = (Base‘𝑅)
86, 7syl6eqr 2808 . . . . 5 (𝑎 = 𝑅 → (Base‘𝑎) = 𝐵)
9 fveq2 6348 . . . . . 6 (𝑏 = 𝑆 → (Base‘𝑏) = (Base‘𝑆))
10 isgim.c . . . . . 6 𝐶 = (Base‘𝑆)
119, 10syl6eqr 2808 . . . . 5 (𝑏 = 𝑆 → (Base‘𝑏) = 𝐶)
12 f1oeq23 6287 . . . . 5 (((Base‘𝑎) = 𝐵 ∧ (Base‘𝑏) = 𝐶) → (𝑐:(Base‘𝑎)–1-1-onto→(Base‘𝑏) ↔ 𝑐:𝐵1-1-onto𝐶))
138, 11, 12syl2an 495 . . . 4 ((𝑎 = 𝑅𝑏 = 𝑆) → (𝑐:(Base‘𝑎)–1-1-onto→(Base‘𝑏) ↔ 𝑐:𝐵1-1-onto𝐶))
145, 13rabeqbidv 3331 . . 3 ((𝑎 = 𝑅𝑏 = 𝑆) → {𝑐 ∈ (𝑎 GrpHom 𝑏) ∣ 𝑐:(Base‘𝑎)–1-1-onto→(Base‘𝑏)} = {𝑐 ∈ (𝑅 GrpHom 𝑆) ∣ 𝑐:𝐵1-1-onto𝐶})
152, 4, 14elovmpt2 7040 . 2 (𝐹 ∈ (𝑅 GrpIso 𝑆) ↔ (𝑅 ∈ Grp ∧ 𝑆 ∈ Grp ∧ 𝐹 ∈ {𝑐 ∈ (𝑅 GrpHom 𝑆) ∣ 𝑐:𝐵1-1-onto𝐶}))
16 ghmgrp1 17859 . . . . . 6 (𝐹 ∈ (𝑅 GrpHom 𝑆) → 𝑅 ∈ Grp)
17 ghmgrp2 17860 . . . . . 6 (𝐹 ∈ (𝑅 GrpHom 𝑆) → 𝑆 ∈ Grp)
1816, 17jca 555 . . . . 5 (𝐹 ∈ (𝑅 GrpHom 𝑆) → (𝑅 ∈ Grp ∧ 𝑆 ∈ Grp))
1918adantr 472 . . . 4 ((𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ 𝐹:𝐵1-1-onto𝐶) → (𝑅 ∈ Grp ∧ 𝑆 ∈ Grp))
2019pm4.71ri 668 . . 3 ((𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ 𝐹:𝐵1-1-onto𝐶) ↔ ((𝑅 ∈ Grp ∧ 𝑆 ∈ Grp) ∧ (𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ 𝐹:𝐵1-1-onto𝐶)))
21 f1oeq1 6284 . . . . 5 (𝑐 = 𝐹 → (𝑐:𝐵1-1-onto𝐶𝐹:𝐵1-1-onto𝐶))
2221elrab 3500 . . . 4 (𝐹 ∈ {𝑐 ∈ (𝑅 GrpHom 𝑆) ∣ 𝑐:𝐵1-1-onto𝐶} ↔ (𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ 𝐹:𝐵1-1-onto𝐶))
2322anbi2i 732 . . 3 (((𝑅 ∈ Grp ∧ 𝑆 ∈ Grp) ∧ 𝐹 ∈ {𝑐 ∈ (𝑅 GrpHom 𝑆) ∣ 𝑐:𝐵1-1-onto𝐶}) ↔ ((𝑅 ∈ Grp ∧ 𝑆 ∈ Grp) ∧ (𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ 𝐹:𝐵1-1-onto𝐶)))
2420, 23bitr4i 267 . 2 ((𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ 𝐹:𝐵1-1-onto𝐶) ↔ ((𝑅 ∈ Grp ∧ 𝑆 ∈ Grp) ∧ 𝐹 ∈ {𝑐 ∈ (𝑅 GrpHom 𝑆) ∣ 𝑐:𝐵1-1-onto𝐶}))
251, 15, 243bitr4i 292 1 (𝐹 ∈ (𝑅 GrpIso 𝑆) ↔ (𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ 𝐹:𝐵1-1-onto𝐶))
Colors of variables: wff setvar class
Syntax hints:  wb 196  wa 383  w3a 1072   = wceq 1628  wcel 2135  {crab 3050  1-1-ontowf1o 6044  cfv 6045  (class class class)co 6809  Basecbs 16055  Grpcgrp 17619   GrpHom cghm 17854   GrpIso cgim 17896
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1867  ax-4 1882  ax-5 1984  ax-6 2050  ax-7 2086  ax-8 2137  ax-9 2144  ax-10 2164  ax-11 2179  ax-12 2192  ax-13 2387  ax-ext 2736  ax-rep 4919  ax-sep 4929  ax-nul 4937  ax-pow 4988  ax-pr 5051  ax-un 7110
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1631  df-ex 1850  df-nf 1855  df-sb 2043  df-eu 2607  df-mo 2608  df-clab 2743  df-cleq 2749  df-clel 2752  df-nfc 2887  df-ne 2929  df-ral 3051  df-rex 3052  df-reu 3053  df-rab 3055  df-v 3338  df-sbc 3573  df-csb 3671  df-dif 3714  df-un 3716  df-in 3718  df-ss 3725  df-nul 4055  df-if 4227  df-pw 4300  df-sn 4318  df-pr 4320  df-op 4324  df-uni 4585  df-iun 4670  df-br 4801  df-opab 4861  df-mpt 4878  df-id 5170  df-xp 5268  df-rel 5269  df-cnv 5270  df-co 5271  df-dm 5272  df-rn 5273  df-res 5274  df-ima 5275  df-iota 6008  df-fun 6047  df-fn 6048  df-f 6049  df-f1 6050  df-fo 6051  df-f1o 6052  df-fv 6053  df-ov 6812  df-oprab 6813  df-mpt2 6814  df-ghm 17855  df-gim 17898
This theorem is referenced by:  gimf1o  17902  gimghm  17903  isgim2  17904  invoppggim  17986  rimgim  18934  lmimgim  19263  zzngim  20099  cygznlem3  20116  pm2mpgrpiso  20820  reefgim  24399  imasgim  38168
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