Theorem grpoinvf 30884

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 7371	. . . 4 ⊢ (℩𝑦 ∈ 𝑋 (𝑦𝐺𝑥) = (GId‘𝐺)) ∈ V
2		eqid 2763	. . . 4 ⊢ (𝑥 ∈ 𝑋 ↦ (℩𝑦 ∈ 𝑋 (𝑦𝐺𝑥) = (GId‘𝐺))) = (𝑥 ∈ 𝑋 ↦ (℩𝑦 ∈ 𝑋 (𝑦𝐺𝑥) = (GId‘𝐺)))
3	1, 2	fnmpti 6678	. . 3 ⊢ (𝑥 ∈ 𝑋 ↦ (℩𝑦 ∈ 𝑋 (𝑦𝐺𝑥) = (GId‘𝐺))) Fn 𝑋
4		grpasscan1.1	. . . . 5 ⊢ 𝑋 = ran 𝐺
5		eqid 2763	. . . . 5 ⊢ (GId‘𝐺) = (GId‘𝐺)
6		grpasscan1.2	. . . . 5 ⊢ 𝑁 = (inv‘𝐺)
7	4, 5, 6	grpoinvfval 30874	. . . 4 ⊢ (𝐺 ∈ GrpOp → 𝑁 = (𝑥 ∈ 𝑋 ↦ (℩𝑦 ∈ 𝑋 (𝑦𝐺𝑥) = (GId‘𝐺))))
8	7	fneq1d 6628	. . 3 ⊢ (𝐺 ∈ GrpOp → (𝑁 Fn 𝑋 ↔ (𝑥 ∈ 𝑋 ↦ (℩𝑦 ∈ 𝑋 (𝑦𝐺𝑥) = (GId‘𝐺))) Fn 𝑋))
9	3, 8	mpbiri 261	. 2 ⊢ (𝐺 ∈ GrpOp → 𝑁 Fn 𝑋)
10		fnrnfv 6940	. . . 4 ⊢ (𝑁 Fn 𝑋 → ran 𝑁 = {𝑦 ∣ ∃𝑥 ∈ 𝑋 𝑦 = (𝑁‘𝑥)})
11	9, 10	syl 18	. . 3 ⊢ (𝐺 ∈ GrpOp → ran 𝑁 = {𝑦 ∣ ∃𝑥 ∈ 𝑋 𝑦 = (𝑁‘𝑥)})
12	4, 6	grpoinvcl 30876	. . . . . . 7 ⊢ ((𝐺 ∈ GrpOp ∧ 𝑦 ∈ 𝑋) → (𝑁‘𝑦) ∈ 𝑋)
13	4, 6	grpo2inv 30883	. . . . . . . 8 ⊢ ((𝐺 ∈ GrpOp ∧ 𝑦 ∈ 𝑋) → (𝑁‘(𝑁‘𝑦)) = 𝑦)
14	13	eqcomd 2769	. . . . . . 7 ⊢ ((𝐺 ∈ GrpOp ∧ 𝑦 ∈ 𝑋) → 𝑦 = (𝑁‘(𝑁‘𝑦)))
15		fveq2 6881	. . . . . . . 8 ⊢ (𝑥 = (𝑁‘𝑦) → (𝑁‘𝑥) = (𝑁‘(𝑁‘𝑦)))
16	15	rspceeqv 3604	. . . . . . 7 ⊢ (((𝑁‘𝑦) ∈ 𝑋 ∧ 𝑦 = (𝑁‘(𝑁‘𝑦))) → ∃𝑥 ∈ 𝑋 𝑦 = (𝑁‘𝑥))
17	12, 14, 16	syl2anc 595	. . . . . 6 ⊢ ((𝐺 ∈ GrpOp ∧ 𝑦 ∈ 𝑋) → ∃𝑥 ∈ 𝑋 𝑦 = (𝑁‘𝑥))
18	17	ex 417	. . . . 5 ⊢ (𝐺 ∈ GrpOp → (𝑦 ∈ 𝑋 → ∃𝑥 ∈ 𝑋 𝑦 = (𝑁‘𝑥)))
19		simpr 489	. . . . . . 7 ⊢ (((𝐺 ∈ GrpOp ∧ 𝑥 ∈ 𝑋) ∧ 𝑦 = (𝑁‘𝑥)) → 𝑦 = (𝑁‘𝑥))
20	4, 6	grpoinvcl 30876	. . . . . . . 8 ⊢ ((𝐺 ∈ GrpOp ∧ 𝑥 ∈ 𝑋) → (𝑁‘𝑥) ∈ 𝑋)
21	20	adantr 485	. . . . . . 7 ⊢ (((𝐺 ∈ GrpOp ∧ 𝑥 ∈ 𝑋) ∧ 𝑦 = (𝑁‘𝑥)) → (𝑁‘𝑥) ∈ 𝑋)
22	19, 21	eqeltrd 2863	. . . . . 6 ⊢ (((𝐺 ∈ GrpOp ∧ 𝑥 ∈ 𝑋) ∧ 𝑦 = (𝑁‘𝑥)) → 𝑦 ∈ 𝑋)
23	22	rexlimdva2 3168	. . . . 5 ⊢ (𝐺 ∈ GrpOp → (∃𝑥 ∈ 𝑋 𝑦 = (𝑁‘𝑥) → 𝑦 ∈ 𝑋))
24	18, 23	impbid 215	. . . 4 ⊢ (𝐺 ∈ GrpOp → (𝑦 ∈ 𝑋 ↔ ∃𝑥 ∈ 𝑋 𝑦 = (𝑁‘𝑥)))
25	24	eqabdv 2896	. . 3 ⊢ (𝐺 ∈ GrpOp → 𝑋 = {𝑦 ∣ ∃𝑥 ∈ 𝑋 𝑦 = (𝑁‘𝑥)})
26	11, 25	eqtr4d 2801	. 2 ⊢ (𝐺 ∈ GrpOp → ran 𝑁 = 𝑋)
27		fveq2 6881	. . . 4 ⊢ ((𝑁‘𝑥) = (𝑁‘𝑦) → (𝑁‘(𝑁‘𝑥)) = (𝑁‘(𝑁‘𝑦)))
28	4, 6	grpo2inv 30883	. . . . . 6 ⊢ ((𝐺 ∈ GrpOp ∧ 𝑥 ∈ 𝑋) → (𝑁‘(𝑁‘𝑥)) = 𝑥)
29	28, 13	eqeqan12d 2777	. . . . 5 ⊢ (((𝐺 ∈ GrpOp ∧ 𝑥 ∈ 𝑋) ∧ (𝐺 ∈ GrpOp ∧ 𝑦 ∈ 𝑋)) → ((𝑁‘(𝑁‘𝑥)) = (𝑁‘(𝑁‘𝑦)) ↔ 𝑥 = 𝑦))
30	29	anandis 690	. . . 4 ⊢ ((𝐺 ∈ GrpOp ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → ((𝑁‘(𝑁‘𝑥)) = (𝑁‘(𝑁‘𝑦)) ↔ 𝑥 = 𝑦))
31	27, 30	imbitrid 247	. . 3 ⊢ ((𝐺 ∈ GrpOp ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → ((𝑁‘𝑥) = (𝑁‘𝑦) → 𝑥 = 𝑦))
32	31	ralrimivva 3208	. 2 ⊢ (𝐺 ∈ GrpOp → ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ((𝑁‘𝑥) = (𝑁‘𝑦) → 𝑥 = 𝑦))
33		dff1o6 7273	. 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 1362	1 ⊢ (𝐺 ∈ GrpOp → 𝑁:𝑋–1-1-onto→𝑋)

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   = wceq 1570   ∈ wcel 2143  {cab 2741  ∀wral 3079  ∃wrex 3089   ↦ cmpt 5192  ran crn 5662   Fn wfn 6531  –1-1-onto→wf1o 6535  ‘cfv 6536  ℩crio 7366  (class class class)co 7410  GrpOpcgr 30841  GIdcgi 30842  invcgn 30843

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-rep 5238  ax-sep 5257  ax-nul 5269  ax-pr 5404  ax-un 7732

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-reu 3370  df-rab 3417  df-v 3457  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4488  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4873  df-iun 4958  df-br 5110  df-opab 5174  df-mpt 5193  df-id 5556  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-res 5673  df-ima 5674  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-riota 7367  df-ov 7413  df-grpo 30845  df-gid 30846  df-ginv 30847

This theorem is referenced by:  nvinvfval  30992