Theorem isgim 19508

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⟶1-1onto𝐶))

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⟶1-1onto𝐶}) ↔ ((𝑅 ∈ Grp ∧ 𝑆 ∈ Grp) ∧ 𝐹 ∈ {𝑐 ∈ (𝑅 GrpHom 𝑆) ∣ 𝑐:𝐵⟶1-1onto𝐶}))
2		df-gim 19505	. . 3 ⊢ GrpIso = (𝑎 ∈ Grp, 𝑏 ∈ Grp ↦ {𝑐 ∈ (𝑎 GrpHom 𝑏) ∣ 𝑐:(Base‘𝑎)⟶1-1onto(Base‘𝑏)})
3		ovex 7479	. . . 4 ⊢ (𝑎 GrpHom 𝑏) ∈ V
4	3	rabex 5305	. . 3 ⊢ {𝑐 ∈ (𝑎 GrpHom 𝑏) ∣ 𝑐:(Base‘𝑎)⟶1-1onto(Base‘𝑏)} ∈ V
5		oveq12 7455	. . . 4 ⊢ ((𝑎 = 𝑅 ∧ 𝑏 = 𝑆) → (𝑎 GrpHom 𝑏) = (𝑅 GrpHom 𝑆))
6		fveq2 6911	. . . . . 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 6911	. . . . . 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 6840	. . . . 5 ⊢ (((Base‘𝑎) = 𝐵 ∧ (Base‘𝑏) = 𝐶) → (𝑐:(Base‘𝑎)⟶1-1onto(Base‘𝑏) ↔ 𝑐:𝐵⟶1-1onto𝐶))
13	8, 11, 12	syl2an 608	. . . 4 ⊢ ((𝑎 = 𝑅 ∧ 𝑏 = 𝑆) → (𝑐:(Base‘𝑎)⟶1-1onto(Base‘𝑏) ↔ 𝑐:𝐵⟶1-1onto𝐶))
14	5, 13	rabeqbidv 3433	. . 3 ⊢ ((𝑎 = 𝑅 ∧ 𝑏 = 𝑆) → {𝑐 ∈ (𝑎 GrpHom 𝑏) ∣ 𝑐:(Base‘𝑎)⟶1-1onto(Base‘𝑏)} = {𝑐 ∈ (𝑅 GrpHom 𝑆) ∣ 𝑐:𝐵⟶1-1onto𝐶})
15	2, 4, 14	elovmpo 7693	. 2 ⊢ (𝐹 ∈ (𝑅 GrpIso 𝑆) ↔ (𝑅 ∈ Grp ∧ 𝑆 ∈ Grp ∧ 𝐹 ∈ {𝑐 ∈ (𝑅 GrpHom 𝑆) ∣ 𝑐:𝐵⟶1-1onto𝐶}))
16		ghmgrp1 19464	. . . . . 6 ⊢ (𝐹 ∈ (𝑅 GrpHom 𝑆) → 𝑅 ∈ Grp)
17		ghmgrp2 19465	. . . . . 6 ⊢ (𝐹 ∈ (𝑅 GrpHom 𝑆) → 𝑆 ∈ Grp)
18	16, 17	jca 521	. . . . 5 ⊢ (𝐹 ∈ (𝑅 GrpHom 𝑆) → (𝑅 ∈ Grp ∧ 𝑆 ∈ Grp))
19	18	adantr 486	. . . 4 ⊢ ((𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ 𝐹:𝐵⟶1-1onto𝐶) → (𝑅 ∈ Grp ∧ 𝑆 ∈ Grp))
20	19	pm4.71ri 570	. . 3 ⊢ ((𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ 𝐹:𝐵⟶1-1onto𝐶) ↔ ((𝑅 ∈ Grp ∧ 𝑆 ∈ Grp) ∧ (𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ 𝐹:𝐵⟶1-1onto𝐶)))
21		f1oeq1 6837	. . . . 5 ⊢ (𝑐 = 𝐹 → (𝑐:𝐵⟶1-1onto𝐶 ↔ 𝐹:𝐵⟶1-1onto𝐶))
22	21	elrab 3648	. . . 4 ⊢ (𝐹 ∈ {𝑐 ∈ (𝑅 GrpHom 𝑆) ∣ 𝑐:𝐵⟶1-1onto𝐶} ↔ (𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ 𝐹:𝐵⟶1-1onto𝐶))
23	22	anbi2i 635	. . 3 ⊢ (((𝑅 ∈ Grp ∧ 𝑆 ∈ Grp) ∧ 𝐹 ∈ {𝑐 ∈ (𝑅 GrpHom 𝑆) ∣ 𝑐:𝐵⟶1-1onto𝐶}) ↔ ((𝑅 ∈ Grp ∧ 𝑆 ∈ Grp) ∧ (𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ 𝐹:𝐵⟶1-1onto𝐶)))
24	20, 23	bitr4i 281	. 2 ⊢ ((𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ 𝐹:𝐵⟶1-1onto𝐶) ↔ ((𝑅 ∈ Grp ∧ 𝑆 ∈ Grp) ∧ 𝐹 ∈ {𝑐 ∈ (𝑅 GrpHom 𝑆) ∣ 𝑐:𝐵⟶1-1onto𝐶}))
25	1, 15, 24	3bitr4i 306	1 ⊢ (𝐹 ∈ (𝑅 GrpIso 𝑆) ↔ (𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ 𝐹:𝐵⟶1-1onto𝐶))

This proof depends on syntax axioms:   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2147  {crab 3416  ⟶1-1ontowf1o 6560  ‘cfv 6561  (class class class)co 7446  Basecbs 17419  Grpcgrp 19176   GrpHom cghm 19459   GrpIso cgim 19503

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-sep 5253  ax-nul 5264  ax-pow 5332  ax-pr 5396  ax-un 7778

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-iun 4956  df-br 5107  df-opab 5172  df-mpt 5191  df-id 5551  df-xp 5662  df-rel 5663  df-cnv 5664  df-co 5665  df-dm 5666  df-rn 5667  df-res 5668  df-ima 5669  df-iota 6515  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7449  df-oprab 7450  df-mpo 7451  df-1st 8028  df-2nd 8029  df-map 8880  df-ghm 19460  df-gim 19505

This theorem is used by:  gimf1o  19509  gimghm  19510  isgim2  19511  imasgim  19516  ghmqusker  19534  invoppggim  19607  rimgim  20767  lmimgim  21373  zzngim  21891  cygznlem3  21908  pm2mpgrpiso  23168  reefgim  26810  grpgimcanon  46134