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Theorem isgim 19280
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 1089 . 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 19277 . . 3 GrpIso = (𝑎 ∈ Grp, 𝑏 ∈ Grp ↦ {𝑐 ∈ (𝑎 GrpHom 𝑏) ∣ 𝑐:(Base‘𝑎)–1-1-onto→(Base‘𝑏)})
3 ovex 7464 . . . 4 (𝑎 GrpHom 𝑏) ∈ V
43rabex 5339 . . 3 {𝑐 ∈ (𝑎 GrpHom 𝑏) ∣ 𝑐:(Base‘𝑎)–1-1-onto→(Base‘𝑏)} ∈ V
5 oveq12 7440 . . . 4 ((𝑎 = 𝑅𝑏 = 𝑆) → (𝑎 GrpHom 𝑏) = (𝑅 GrpHom 𝑆))
6 fveq2 6906 . . . . . 6 (𝑎 = 𝑅 → (Base‘𝑎) = (Base‘𝑅))
7 isgim.b . . . . . 6 𝐵 = (Base‘𝑅)
86, 7eqtr4di 2795 . . . . 5 (𝑎 = 𝑅 → (Base‘𝑎) = 𝐵)
9 fveq2 6906 . . . . . 6 (𝑏 = 𝑆 → (Base‘𝑏) = (Base‘𝑆))
10 isgim.c . . . . . 6 𝐶 = (Base‘𝑆)
119, 10eqtr4di 2795 . . . . 5 (𝑏 = 𝑆 → (Base‘𝑏) = 𝐶)
12 f1oeq23 6839 . . . . 5 (((Base‘𝑎) = 𝐵 ∧ (Base‘𝑏) = 𝐶) → (𝑐:(Base‘𝑎)–1-1-onto→(Base‘𝑏) ↔ 𝑐:𝐵1-1-onto𝐶))
138, 11, 12syl2an 596 . . . 4 ((𝑎 = 𝑅𝑏 = 𝑆) → (𝑐:(Base‘𝑎)–1-1-onto→(Base‘𝑏) ↔ 𝑐:𝐵1-1-onto𝐶))
145, 13rabeqbidv 3455 . . 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, 14elovmpo 7678 . 2 (𝐹 ∈ (𝑅 GrpIso 𝑆) ↔ (𝑅 ∈ Grp ∧ 𝑆 ∈ Grp ∧ 𝐹 ∈ {𝑐 ∈ (𝑅 GrpHom 𝑆) ∣ 𝑐:𝐵1-1-onto𝐶}))
16 ghmgrp1 19236 . . . . . 6 (𝐹 ∈ (𝑅 GrpHom 𝑆) → 𝑅 ∈ Grp)
17 ghmgrp2 19237 . . . . . 6 (𝐹 ∈ (𝑅 GrpHom 𝑆) → 𝑆 ∈ Grp)
1816, 17jca 511 . . . . 5 (𝐹 ∈ (𝑅 GrpHom 𝑆) → (𝑅 ∈ Grp ∧ 𝑆 ∈ Grp))
1918adantr 480 . . . 4 ((𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ 𝐹:𝐵1-1-onto𝐶) → (𝑅 ∈ Grp ∧ 𝑆 ∈ Grp))
2019pm4.71ri 560 . . 3 ((𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ 𝐹:𝐵1-1-onto𝐶) ↔ ((𝑅 ∈ Grp ∧ 𝑆 ∈ Grp) ∧ (𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ 𝐹:𝐵1-1-onto𝐶)))
21 f1oeq1 6836 . . . . 5 (𝑐 = 𝐹 → (𝑐:𝐵1-1-onto𝐶𝐹:𝐵1-1-onto𝐶))
2221elrab 3692 . . . 4 (𝐹 ∈ {𝑐 ∈ (𝑅 GrpHom 𝑆) ∣ 𝑐:𝐵1-1-onto𝐶} ↔ (𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ 𝐹:𝐵1-1-onto𝐶))
2322anbi2i 623 . . 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 278 . 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 303 1 (𝐹 ∈ (𝑅 GrpIso 𝑆) ↔ (𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ 𝐹:𝐵1-1-onto𝐶))
Colors of variables: wff setvar class
Syntax hints:  wb 206  wa 395  w3a 1087   = wceq 1540  wcel 2108  {crab 3436  1-1-ontowf1o 6560  cfv 6561  (class class class)co 7431  Basecbs 17247  Grpcgrp 18951   GrpHom cghm 19230   GrpIso cgim 19275
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8014  df-2nd 8015  df-map 8868  df-ghm 19231  df-gim 19277
This theorem is referenced by:  gimf1o  19281  gimghm  19282  isgim2  19283  ghmqusker  19305  invoppggim  19379  rimgim  20497  lmimgim  21064  zzngim  21571  cygznlem3  21588  pm2mpgrpiso  22823  reefgim  26494  imasgim  43112
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