Theorem grpoinvf 29002

Description: Mapping of the inverse function of a group. (Contributed by NM, 29-Mar-2008.) (Revised by Mario Carneiro, 15-Dec-2013.) (New usage is discouraged.)

Hypotheses

Ref	Expression
grpasscan1.1	⊢ 𝑋 = ran 𝐺
grpasscan1.2	⊢ 𝑁 = (inv‘𝐺)

Assertion

Ref	Expression
grpoinvf	⊢ (𝐺 ∈ GrpOp → 𝑁:𝑋–1-1-onto→𝑋)

Proof of Theorem grpoinvf

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		riotaex 7274	. . . 4 ⊢ (℩𝑦 ∈ 𝑋 (𝑦𝐺𝑥) = (GId‘𝐺)) ∈ V
2		eqid 2737	. . . 4 ⊢ (𝑥 ∈ 𝑋 ↦ (℩𝑦 ∈ 𝑋 (𝑦𝐺𝑥) = (GId‘𝐺))) = (𝑥 ∈ 𝑋 ↦ (℩𝑦 ∈ 𝑋 (𝑦𝐺𝑥) = (GId‘𝐺)))
3	1, 2	fnmpti 6611	. . 3 ⊢ (𝑥 ∈ 𝑋 ↦ (℩𝑦 ∈ 𝑋 (𝑦𝐺𝑥) = (GId‘𝐺))) Fn 𝑋
4		grpasscan1.1	. . . . 5 ⊢ 𝑋 = ran 𝐺
5		eqid 2737	. . . . 5 ⊢ (GId‘𝐺) = (GId‘𝐺)
6		grpasscan1.2	. . . . 5 ⊢ 𝑁 = (inv‘𝐺)
7	4, 5, 6	grpoinvfval 28992	. . . 4 ⊢ (𝐺 ∈ GrpOp → 𝑁 = (𝑥 ∈ 𝑋 ↦ (℩𝑦 ∈ 𝑋 (𝑦𝐺𝑥) = (GId‘𝐺))))
8	7	fneq1d 6562	. . 3 ⊢ (𝐺 ∈ GrpOp → (𝑁 Fn 𝑋 ↔ (𝑥 ∈ 𝑋 ↦ (℩𝑦 ∈ 𝑋 (𝑦𝐺𝑥) = (GId‘𝐺))) Fn 𝑋))
9	3, 8	mpbiri 257	. 2 ⊢ (𝐺 ∈ GrpOp → 𝑁 Fn 𝑋)
10		fnrnfv 6866	. . . 4 ⊢ (𝑁 Fn 𝑋 → ran 𝑁 = {𝑦 ∣ ∃𝑥 ∈ 𝑋 𝑦 = (𝑁‘𝑥)})
11	9, 10	syl 17	. . 3 ⊢ (𝐺 ∈ GrpOp → ran 𝑁 = {𝑦 ∣ ∃𝑥 ∈ 𝑋 𝑦 = (𝑁‘𝑥)})
12	4, 6	grpoinvcl 28994	. . . . . . 7 ⊢ ((𝐺 ∈ GrpOp ∧ 𝑦 ∈ 𝑋) → (𝑁‘𝑦) ∈ 𝑋)
13	4, 6	grpo2inv 29001	. . . . . . . 8 ⊢ ((𝐺 ∈ GrpOp ∧ 𝑦 ∈ 𝑋) → (𝑁‘(𝑁‘𝑦)) = 𝑦)
14	13	eqcomd 2743	. . . . . . 7 ⊢ ((𝐺 ∈ GrpOp ∧ 𝑦 ∈ 𝑋) → 𝑦 = (𝑁‘(𝑁‘𝑦)))
15		fveq2 6809	. . . . . . . 8 ⊢ (𝑥 = (𝑁‘𝑦) → (𝑁‘𝑥) = (𝑁‘(𝑁‘𝑦)))
16	15	rspceeqv 3584	. . . . . . 7 ⊢ (((𝑁‘𝑦) ∈ 𝑋 ∧ 𝑦 = (𝑁‘(𝑁‘𝑦))) → ∃𝑥 ∈ 𝑋 𝑦 = (𝑁‘𝑥))
17	12, 14, 16	syl2anc 584	. . . . . 6 ⊢ ((𝐺 ∈ GrpOp ∧ 𝑦 ∈ 𝑋) → ∃𝑥 ∈ 𝑋 𝑦 = (𝑁‘𝑥))
18	17	ex 413	. . . . 5 ⊢ (𝐺 ∈ GrpOp → (𝑦 ∈ 𝑋 → ∃𝑥 ∈ 𝑋 𝑦 = (𝑁‘𝑥)))
19		simpr 485	. . . . . . 7 ⊢ (((𝐺 ∈ GrpOp ∧ 𝑥 ∈ 𝑋) ∧ 𝑦 = (𝑁‘𝑥)) → 𝑦 = (𝑁‘𝑥))
20	4, 6	grpoinvcl 28994	. . . . . . . 8 ⊢ ((𝐺 ∈ GrpOp ∧ 𝑥 ∈ 𝑋) → (𝑁‘𝑥) ∈ 𝑋)
21	20	adantr 481	. . . . . . 7 ⊢ (((𝐺 ∈ GrpOp ∧ 𝑥 ∈ 𝑋) ∧ 𝑦 = (𝑁‘𝑥)) → (𝑁‘𝑥) ∈ 𝑋)
22	19, 21	eqeltrd 2838	. . . . . 6 ⊢ (((𝐺 ∈ GrpOp ∧ 𝑥 ∈ 𝑋) ∧ 𝑦 = (𝑁‘𝑥)) → 𝑦 ∈ 𝑋)
23	22	rexlimdva2 3151	. . . . 5 ⊢ (𝐺 ∈ GrpOp → (∃𝑥 ∈ 𝑋 𝑦 = (𝑁‘𝑥) → 𝑦 ∈ 𝑋))
24	18, 23	impbid 211	. . . 4 ⊢ (𝐺 ∈ GrpOp → (𝑦 ∈ 𝑋 ↔ ∃𝑥 ∈ 𝑋 𝑦 = (𝑁‘𝑥)))
25	24	abbi2dv 2876	. . 3 ⊢ (𝐺 ∈ GrpOp → 𝑋 = {𝑦 ∣ ∃𝑥 ∈ 𝑋 𝑦 = (𝑁‘𝑥)})
26	11, 25	eqtr4d 2780	. 2 ⊢ (𝐺 ∈ GrpOp → ran 𝑁 = 𝑋)
27		fveq2 6809	. . . 4 ⊢ ((𝑁‘𝑥) = (𝑁‘𝑦) → (𝑁‘(𝑁‘𝑥)) = (𝑁‘(𝑁‘𝑦)))
28	4, 6	grpo2inv 29001	. . . . . 6 ⊢ ((𝐺 ∈ GrpOp ∧ 𝑥 ∈ 𝑋) → (𝑁‘(𝑁‘𝑥)) = 𝑥)
29	28, 13	eqeqan12d 2751	. . . . 5 ⊢ (((𝐺 ∈ GrpOp ∧ 𝑥 ∈ 𝑋) ∧ (𝐺 ∈ GrpOp ∧ 𝑦 ∈ 𝑋)) → ((𝑁‘(𝑁‘𝑥)) = (𝑁‘(𝑁‘𝑦)) ↔ 𝑥 = 𝑦))
30	29	anandis 675	. . . 4 ⊢ ((𝐺 ∈ GrpOp ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → ((𝑁‘(𝑁‘𝑥)) = (𝑁‘(𝑁‘𝑦)) ↔ 𝑥 = 𝑦))
31	27, 30	syl5ib 243	. . 3 ⊢ ((𝐺 ∈ GrpOp ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → ((𝑁‘𝑥) = (𝑁‘𝑦) → 𝑥 = 𝑦))
32	31	ralrimivva 3194	. 2 ⊢ (𝐺 ∈ GrpOp → ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ((𝑁‘𝑥) = (𝑁‘𝑦) → 𝑥 = 𝑦))
33		dff1o6 7184	. 2 ⊢ (𝑁:𝑋–1-1-onto→𝑋 ↔ (𝑁 Fn 𝑋 ∧ ran 𝑁 = 𝑋 ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ((𝑁‘𝑥) = (𝑁‘𝑦) → 𝑥 = 𝑦)))
34	9, 26, 32, 33	syl3anbrc 1342	1 ⊢ (𝐺 ∈ GrpOp → 𝑁:𝑋–1-1-onto→𝑋)

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   = wceq 1540   ∈ wcel 2105  {cab 2714  ∀wral 3062  ∃wrex 3071   ↦ cmpt 5168  ran crn 5606   Fn wfn 6458  –1-1-onto→wf1o 6462  ‘cfv 6463  ℩crio 7269  (class class class)co 7313  GrpOpcgr 28959  GIdcgi 28960  invcgn 28961

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 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2708  ax-rep 5222  ax-sep 5236  ax-nul 5243  ax-pr 5365  ax-un 7626

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2887  df-ne 2942  df-ral 3063  df-rex 3072  df-reu 3351  df-rab 3405  df-v 3443  df-sbc 3726  df-csb 3842  df-dif 3899  df-un 3901  df-in 3903  df-ss 3913  df-nul 4267  df-if 4470  df-sn 4570  df-pr 4572  df-op 4576  df-uni 4849  df-iun 4937  df-br 5086  df-opab 5148  df-mpt 5169  df-id 5505  df-xp 5611  df-rel 5612  df-cnv 5613  df-co 5614  df-dm 5615  df-rn 5616  df-res 5617  df-ima 5618  df-iota 6415  df-fun 6465  df-fn 6466  df-f 6467  df-f1 6468  df-fo 6469  df-f1o 6470  df-fv 6471  df-riota 7270  df-ov 7316  df-grpo 28963  df-gid 28964  df-ginv 28965

This theorem is referenced by:  nvinvfval  29110