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: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 1099	. 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 19289	. . 3 ⊢ GrpIso = (𝑎 ∈ Grp, 𝑏 ∈ Grp ↦ {𝑐 ∈ (𝑎 GrpHom 𝑏) ∣ 𝑐:(Base‘𝑎)–1-1-onto→(Base‘𝑏)})
3		ovex 7423	. . . 4 ⊢ (𝑎 GrpHom 𝑏) ∈ V
4	3	rabex 5292	. . 3 ⊢ {𝑐 ∈ (𝑎 GrpHom 𝑏) ∣ 𝑐:(Base‘𝑎)–1-1-onto→(Base‘𝑏)} ∈ V
5		oveq12 7399	. . . 4 ⊢ ((𝑎 = 𝑅 ∧ 𝑏 = 𝑆) → (𝑎 GrpHom 𝑏) = (𝑅 GrpHom 𝑆))
6		fveq2 6861	. . . . . 6 ⊢ (𝑎 = 𝑅 → (Base‘𝑎) = (Base‘𝑅))
7		isgim.b	. . . . . 6 ⊢ 𝐵 = (Base‘𝑅)
8	6, 7	eqtr4di 2814	. . . . 5 ⊢ (𝑎 = 𝑅 → (Base‘𝑎) = 𝐵)
9		fveq2 6861	. . . . . 6 ⊢ (𝑏 = 𝑆 → (Base‘𝑏) = (Base‘𝑆))
10		isgim.c	. . . . . 6 ⊢ 𝐶 = (Base‘𝑆)
11	9, 10	eqtr4di 2814	. . . . 5 ⊢ (𝑏 = 𝑆 → (Base‘𝑏) = 𝐶)
12		f1oeq23 6791	. . . . 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 605	. . . 4 ⊢ ((𝑎 = 𝑅 ∧ 𝑏 = 𝑆) → (𝑐:(Base‘𝑎)–1-1-onto→(Base‘𝑏) ↔ 𝑐:𝐵–1-1-onto→𝐶))
14	5, 13	rabeqbidv 3431	. . 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 7635	. 2 ⊢ (𝐹 ∈ (𝑅 GrpIso 𝑆) ↔ (𝑅 ∈ Grp ∧ 𝑆 ∈ Grp ∧ 𝐹 ∈ {𝑐 ∈ (𝑅 GrpHom 𝑆) ∣ 𝑐:𝐵–1-1-onto→𝐶}))
16		ghmgrp1 19248	. . . . . 6 ⊢ (𝐹 ∈ (𝑅 GrpHom 𝑆) → 𝑅 ∈ Grp)
17		ghmgrp2 19249	. . . . . 6 ⊢ (𝐹 ∈ (𝑅 GrpHom 𝑆) → 𝑆 ∈ Grp)
18	16, 17	jca 519	. . . . 5 ⊢ (𝐹 ∈ (𝑅 GrpHom 𝑆) → (𝑅 ∈ Grp ∧ 𝑆 ∈ Grp))
19	18	adantr 484	. . . 4 ⊢ ((𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ 𝐹:𝐵–1-1-onto→𝐶) → (𝑅 ∈ Grp ∧ 𝑆 ∈ Grp))
20	19	pm4.71ri 568	. . 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 6788	. . . . 5 ⊢ (𝑐 = 𝐹 → (𝑐:𝐵–1-1-onto→𝐶 ↔ 𝐹:𝐵–1-1-onto→𝐶))
22	21	elrab 3649	. . . 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 632	. . 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 399   ∧ w3a 1097   = wceq 1559   ∈ wcel 2141  {crab 3413  –1-1-onto→wf1o 6514  ‘cfv 6515  (class class class)co 7390  Basecbs 17235  Grpcgrp 18965   GrpHom cghm 19243   GrpIso cgim 19287

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-sep 5243  ax-nul 5253  ax-pow 5319  ax-pr 5387  ax-un 7712

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3076  df-rex 3086  df-rab 3414  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4284  df-if 4478  df-pw 4554  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4863  df-iun 4948  df-br 5098  df-opab 5160  df-mpt 5179  df-id 5538  df-xp 5649  df-rel 5650  df-cnv 5651  df-co 5652  df-dm 5653  df-rn 5654  df-res 5655  df-ima 5656  df-iota 6471  df-fun 6517  df-fn 6518  df-f 6519  df-f1 6520  df-fo 6521  df-f1o 6522  df-fv 6523  df-ov 7393  df-oprab 7394  df-mpo 7395  df-1st 7964  df-2nd 7965  df-map 8803  df-ghm 19244  df-gim 19289

This theorem is referenced by:  gimf1o  19293  gimghm  19294  isgim2  19295  ghmqusker  19317  invoppggim  19390  rimgim  20532  lmimgim  21119  zzngim  21591  cygznlem3  21608  pm2mpgrpiso  22864  reefgim  26500  imasgim  43637