Theorem isgim 19333

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 1105	. 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 19330	. . 3 ⊢ GrpIso = (𝑎 ∈ Grp, 𝑏 ∈ Grp ↦ {𝑐 ∈ (𝑎 GrpHom 𝑏) ∣ 𝑐:(Base‘𝑎)–1-1-onto→(Base‘𝑏)})
3		ovex 7445	. . . 4 ⊢ (𝑎 GrpHom 𝑏) ∈ V
4	3	rabex 5311	. . 3 ⊢ {𝑐 ∈ (𝑎 GrpHom 𝑏) ∣ 𝑐:(Base‘𝑎)–1-1-onto→(Base‘𝑏)} ∈ V
5		oveq12 7421	. . . 4 ⊢ ((𝑎 = 𝑅 ∧ 𝑏 = 𝑆) → (𝑎 GrpHom 𝑏) = (𝑅 GrpHom 𝑆))
6		fveq2 6883	. . . . . 6 ⊢ (𝑎 = 𝑅 → (Base‘𝑎) = (Base‘𝑅))
7		isgim.b	. . . . . 6 ⊢ 𝐵 = (Base‘𝑅)
8	6, 7	eqtr4di 2816	. . . . 5 ⊢ (𝑎 = 𝑅 → (Base‘𝑎) = 𝐵)
9		fveq2 6883	. . . . . 6 ⊢ (𝑏 = 𝑆 → (Base‘𝑏) = (Base‘𝑆))
10		isgim.c	. . . . . 6 ⊢ 𝐶 = (Base‘𝑆)
11	9, 10	eqtr4di 2816	. . . . 5 ⊢ (𝑏 = 𝑆 → (Base‘𝑏) = 𝐶)
12		f1oeq23 6813	. . . . 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 607	. . . 4 ⊢ ((𝑎 = 𝑅 ∧ 𝑏 = 𝑆) → (𝑐:(Base‘𝑎)–1-1-onto→(Base‘𝑏) ↔ 𝑐:𝐵–1-1-onto→𝐶))
14	5, 13	rabeqbidv 3434	. . 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 7657	. 2 ⊢ (𝐹 ∈ (𝑅 GrpIso 𝑆) ↔ (𝑅 ∈ Grp ∧ 𝑆 ∈ Grp ∧ 𝐹 ∈ {𝑐 ∈ (𝑅 GrpHom 𝑆) ∣ 𝑐:𝐵–1-1-onto→𝐶}))
16		ghmgrp1 19289	. . . . . 6 ⊢ (𝐹 ∈ (𝑅 GrpHom 𝑆) → 𝑅 ∈ Grp)
17		ghmgrp2 19290	. . . . . 6 ⊢ (𝐹 ∈ (𝑅 GrpHom 𝑆) → 𝑆 ∈ Grp)
18	16, 17	jca 520	. . . . 5 ⊢ (𝐹 ∈ (𝑅 GrpHom 𝑆) → (𝑅 ∈ Grp ∧ 𝑆 ∈ Grp))
19	18	adantr 485	. . . 4 ⊢ ((𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ 𝐹:𝐵–1-1-onto→𝐶) → (𝑅 ∈ Grp ∧ 𝑆 ∈ Grp))
20	19	pm4.71ri 569	. . 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 6810	. . . . 5 ⊢ (𝑐 = 𝐹 → (𝑐:𝐵–1-1-onto→𝐶 ↔ 𝐹:𝐵–1-1-onto→𝐶))
22	21	elrab 3651	. . . 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 634	. . 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 281	. 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 306	1 ⊢ (𝐹 ∈ (𝑅 GrpIso 𝑆) ↔ (𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ 𝐹:𝐵–1-1-onto→𝐶))

Syntax hints:   ↔ wb 209   ∧ wa 400   ∧ w3a 1103   = wceq 1570   ∈ wcel 2143  {crab 3416  –1-1-onto→wf1o 6537  ‘cfv 6538  (class class class)co 7412  Basecbs 17270  Grpcgrp 19001   GrpHom cghm 19284   GrpIso cgim 19328

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-sep 5258  ax-nul 5270  ax-pow 5338  ax-pr 5406  ax-un 7734

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-rab 3417  df-v 3457  df-sbc 3746  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4288  df-if 4489  df-pw 4565  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4874  df-iun 4959  df-br 5111  df-opab 5175  df-mpt 5194  df-id 5558  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-ov 7415  df-oprab 7416  df-mpo 7417  df-1st 7987  df-2nd 7988  df-map 8827  df-ghm 19285  df-gim 19330

This theorem is referenced by:  gimf1o  19334  gimghm  19335  isgim2  19336  ghmqusker  19358  invoppggim  19431  rimgim  20580  lmimgim  21167  zzngim  21683  cygznlem3  21700  pm2mpgrpiso  22955  reefgim  26594  imasgim  43810