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Theorem isgim 19292
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 1088 . 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 19289 . . 3 GrpIso = (𝑎 ∈ Grp, 𝑏 ∈ Grp ↦ {𝑐 ∈ (𝑎 GrpHom 𝑏) ∣ 𝑐:(Base‘𝑎)–1-1-onto→(Base‘𝑏)})
3 ovex 7463 . . . 4 (𝑎 GrpHom 𝑏) ∈ V
43rabex 5344 . . 3 {𝑐 ∈ (𝑎 GrpHom 𝑏) ∣ 𝑐:(Base‘𝑎)–1-1-onto→(Base‘𝑏)} ∈ V
5 oveq12 7439 . . . 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 2792 . . . . 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 2792 . . . . 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 3451 . . 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 7677 . 2 (𝐹 ∈ (𝑅 GrpIso 𝑆) ↔ (𝑅 ∈ Grp ∧ 𝑆 ∈ Grp ∧ 𝐹 ∈ {𝑐 ∈ (𝑅 GrpHom 𝑆) ∣ 𝑐:𝐵1-1-onto𝐶}))
16 ghmgrp1 19248 . . . . . 6 (𝐹 ∈ (𝑅 GrpHom 𝑆) → 𝑅 ∈ Grp)
17 ghmgrp2 19249 . . . . . 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 3694 . . . 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 1086   = wceq 1536  wcel 2105  {crab 3432  1-1-ontowf1o 6561  cfv 6562  (class class class)co 7430  Basecbs 17244  Grpcgrp 18963   GrpHom cghm 19242   GrpIso cgim 19287
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-sep 5301  ax-nul 5311  ax-pow 5370  ax-pr 5437  ax-un 7753
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-iun 4997  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5582  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-f1 6567  df-fo 6568  df-f1o 6569  df-fv 6570  df-ov 7433  df-oprab 7434  df-mpo 7435  df-1st 8012  df-2nd 8013  df-map 8866  df-ghm 19243  df-gim 19289
This theorem is referenced by:  gimf1o  19293  gimghm  19294  isgim2  19295  ghmqusker  19317  invoppggim  19393  rimgim  20513  lmimgim  21081  zzngim  21588  cygznlem3  21605  pm2mpgrpiso  22838  reefgim  26508  imasgim  43088
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