Theorem isgim 18404

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:𝐵–1-1-onto→𝐶))

Proof of Theorem isgim

Dummy variables 𝑎 𝑏 𝑐 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-3an 1085	. 2 ⊢ ((𝑅 ∈ Grp ∧ 𝑆 ∈ Grp ∧ 𝐹 ∈ {𝑐 ∈ (𝑅 GrpHom 𝑆) ∣ 𝑐:𝐵–1-1-onto→𝐶}) ↔ ((𝑅 ∈ Grp ∧ 𝑆 ∈ Grp) ∧ 𝐹 ∈ {𝑐 ∈ (𝑅 GrpHom 𝑆) ∣ 𝑐:𝐵–1-1-onto→𝐶}))
2		df-gim 18401	. . 3 ⊢ GrpIso = (𝑎 ∈ Grp, 𝑏 ∈ Grp ↦ {𝑐 ∈ (𝑎 GrpHom 𝑏) ∣ 𝑐:(Base‘𝑎)–1-1-onto→(Base‘𝑏)})
3		ovex 7191	. . . 4 ⊢ (𝑎 GrpHom 𝑏) ∈ V
4	3	rabex 5237	. . 3 ⊢ {𝑐 ∈ (𝑎 GrpHom 𝑏) ∣ 𝑐:(Base‘𝑎)–1-1-onto→(Base‘𝑏)} ∈ V
5		oveq12 7167	. . . 4 ⊢ ((𝑎 = 𝑅 ∧ 𝑏 = 𝑆) → (𝑎 GrpHom 𝑏) = (𝑅 GrpHom 𝑆))
6		fveq2 6672	. . . . . 6 ⊢ (𝑎 = 𝑅 → (Base‘𝑎) = (Base‘𝑅))
7		isgim.b	. . . . . 6 ⊢ 𝐵 = (Base‘𝑅)
8	6, 7	syl6eqr 2876	. . . . 5 ⊢ (𝑎 = 𝑅 → (Base‘𝑎) = 𝐵)
9		fveq2 6672	. . . . . 6 ⊢ (𝑏 = 𝑆 → (Base‘𝑏) = (Base‘𝑆))
10		isgim.c	. . . . . 6 ⊢ 𝐶 = (Base‘𝑆)
11	9, 10	syl6eqr 2876	. . . . 5 ⊢ (𝑏 = 𝑆 → (Base‘𝑏) = 𝐶)
12		f1oeq23 6609	. . . . 5 ⊢ (((Base‘𝑎) = 𝐵 ∧ (Base‘𝑏) = 𝐶) → (𝑐:(Base‘𝑎)–1-1-onto→(Base‘𝑏) ↔ 𝑐:𝐵–1-1-onto→𝐶))
13	8, 11, 12	syl2an 597	. . . 4 ⊢ ((𝑎 = 𝑅 ∧ 𝑏 = 𝑆) → (𝑐:(Base‘𝑎)–1-1-onto→(Base‘𝑏) ↔ 𝑐:𝐵–1-1-onto→𝐶))
14	5, 13	rabeqbidv 3487	. . 3 ⊢ ((𝑎 = 𝑅 ∧ 𝑏 = 𝑆) → {𝑐 ∈ (𝑎 GrpHom 𝑏) ∣ 𝑐:(Base‘𝑎)–1-1-onto→(Base‘𝑏)} = {𝑐 ∈ (𝑅 GrpHom 𝑆) ∣ 𝑐:𝐵–1-1-onto→𝐶})
15	2, 4, 14	elovmpo 7392	. 2 ⊢ (𝐹 ∈ (𝑅 GrpIso 𝑆) ↔ (𝑅 ∈ Grp ∧ 𝑆 ∈ Grp ∧ 𝐹 ∈ {𝑐 ∈ (𝑅 GrpHom 𝑆) ∣ 𝑐:𝐵–1-1-onto→𝐶}))
16		ghmgrp1 18362	. . . . . 6 ⊢ (𝐹 ∈ (𝑅 GrpHom 𝑆) → 𝑅 ∈ Grp)
17		ghmgrp2 18363	. . . . . 6 ⊢ (𝐹 ∈ (𝑅 GrpHom 𝑆) → 𝑆 ∈ Grp)
18	16, 17	jca 514	. . . . 5 ⊢ (𝐹 ∈ (𝑅 GrpHom 𝑆) → (𝑅 ∈ Grp ∧ 𝑆 ∈ Grp))
19	18	adantr 483	. . . 4 ⊢ ((𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ 𝐹:𝐵–1-1-onto→𝐶) → (𝑅 ∈ Grp ∧ 𝑆 ∈ Grp))
20	19	pm4.71ri 563	. . 3 ⊢ ((𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ 𝐹:𝐵–1-1-onto→𝐶) ↔ ((𝑅 ∈ Grp ∧ 𝑆 ∈ Grp) ∧ (𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ 𝐹:𝐵–1-1-onto→𝐶)))
21		f1oeq1 6606	. . . . 5 ⊢ (𝑐 = 𝐹 → (𝑐:𝐵–1-1-onto→𝐶 ↔ 𝐹:𝐵–1-1-onto→𝐶))
22	21	elrab 3682	. . . 4 ⊢ (𝐹 ∈ {𝑐 ∈ (𝑅 GrpHom 𝑆) ∣ 𝑐:𝐵–1-1-onto→𝐶} ↔ (𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ 𝐹:𝐵–1-1-onto→𝐶))
23	22	anbi2i 624	. . 3 ⊢ (((𝑅 ∈ Grp ∧ 𝑆 ∈ Grp) ∧ 𝐹 ∈ {𝑐 ∈ (𝑅 GrpHom 𝑆) ∣ 𝑐:𝐵–1-1-onto→𝐶}) ↔ ((𝑅 ∈ Grp ∧ 𝑆 ∈ Grp) ∧ (𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ 𝐹:𝐵–1-1-onto→𝐶)))
24	20, 23	bitr4i 280	. 2 ⊢ ((𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ 𝐹:𝐵–1-1-onto→𝐶) ↔ ((𝑅 ∈ Grp ∧ 𝑆 ∈ Grp) ∧ 𝐹 ∈ {𝑐 ∈ (𝑅 GrpHom 𝑆) ∣ 𝑐:𝐵–1-1-onto→𝐶}))
25	1, 15, 24	3bitr4i 305	1 ⊢ (𝐹 ∈ (𝑅 GrpIso 𝑆) ↔ (𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ 𝐹:𝐵–1-1-onto→𝐶))

Syntax hints:   ↔ wb 208   ∧ wa 398   ∧ w3a 1083   = wceq 1537   ∈ wcel 2114  {crab 3144  –1-1-onto→wf1o 6356  ‘cfv 6357  (class class class)co 7158  Basecbs 16485  Grpcgrp 18105   GrpHom cghm 18357   GrpIso cgim 18399

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795  ax-rep 5192  ax-sep 5205  ax-nul 5212  ax-pow 5268  ax-pr 5332  ax-un 7463

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ne 3019  df-ral 3145  df-rex 3146  df-reu 3147  df-rab 3149  df-v 3498  df-sbc 3775  df-csb 3886  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-nul 4294  df-if 4470  df-pw 4543  df-sn 4570  df-pr 4572  df-op 4576  df-uni 4841  df-iun 4923  df-br 5069  df-opab 5131  df-mpt 5149  df-id 5462  df-xp 5563  df-rel 5564  df-cnv 5565  df-co 5566  df-dm 5567  df-rn 5568  df-res 5569  df-ima 5570  df-iota 6316  df-fun 6359  df-fn 6360  df-f 6361  df-f1 6362  df-fo 6363  df-f1o 6364  df-fv 6365  df-ov 7161  df-oprab 7162  df-mpo 7163  df-ghm 18358  df-gim 18401

This theorem is referenced by:  gimf1o  18405  gimghm  18406  isgim2  18407  invoppggim  18490  rimgim  19490  lmimgim  19839  zzngim  20701  cygznlem3  20718  pm2mpgrpiso  21427  reefgim  25040  imasgim  39707