Theorem grpoinvf 30824

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 7372	. . . 4 ⊢ (℩𝑦 ∈ 𝑋 (𝑦𝐺𝑥) = (GId‘𝐺)) ∈ V
2		eqid 2769	. . . 4 ⊢ (𝑥 ∈ 𝑋 ↦ (℩𝑦 ∈ 𝑋 (𝑦𝐺𝑥) = (GId‘𝐺))) = (𝑥 ∈ 𝑋 ↦ (℩𝑦 ∈ 𝑋 (𝑦𝐺𝑥) = (GId‘𝐺)))
3	1, 2	fnmpti 6679	. . 3 ⊢ (𝑥 ∈ 𝑋 ↦ (℩𝑦 ∈ 𝑋 (𝑦𝐺𝑥) = (GId‘𝐺))) Fn 𝑋
4		grpasscan1.1	. . . . 5 ⊢ 𝑋 = ran 𝐺
5		eqid 2769	. . . . 5 ⊢ (GId‘𝐺) = (GId‘𝐺)
6		grpasscan1.2	. . . . 5 ⊢ 𝑁 = (inv‘𝐺)
7	4, 5, 6	grpoinvfval 30814	. . . 4 ⊢ (𝐺 ∈ GrpOp → 𝑁 = (𝑥 ∈ 𝑋 ↦ (℩𝑦 ∈ 𝑋 (𝑦𝐺𝑥) = (GId‘𝐺))))
8	7	fneq1d 6629	. . 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 6941	. . . 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 30816	. . . . . . 7 ⊢ ((𝐺 ∈ GrpOp ∧ 𝑦 ∈ 𝑋) → (𝑁‘𝑦) ∈ 𝑋)
13	4, 6	grpo2inv 30823	. . . . . . . 8 ⊢ ((𝐺 ∈ GrpOp ∧ 𝑦 ∈ 𝑋) → (𝑁‘(𝑁‘𝑦)) = 𝑦)
14	13	eqcomd 2775	. . . . . . 7 ⊢ ((𝐺 ∈ GrpOp ∧ 𝑦 ∈ 𝑋) → 𝑦 = (𝑁‘(𝑁‘𝑦)))
15		fveq2 6882	. . . . . . . 8 ⊢ (𝑥 = (𝑁‘𝑦) → (𝑁‘𝑥) = (𝑁‘(𝑁‘𝑦)))
16	15	rspceeqv 3613	. . . . . . 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 30816	. . . . . . . 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 2869	. . . . . 6 ⊢ (((𝐺 ∈ GrpOp ∧ 𝑥 ∈ 𝑋) ∧ 𝑦 = (𝑁‘𝑥)) → 𝑦 ∈ 𝑋)
23	22	rexlimdva2 3174	. . . . 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 2902	. . 3 ⊢ (𝐺 ∈ GrpOp → 𝑋 = {𝑦 ∣ ∃𝑥 ∈ 𝑋 𝑦 = (𝑁‘𝑥)})
26	11, 25	eqtr4d 2807	. 2 ⊢ (𝐺 ∈ GrpOp → ran 𝑁 = 𝑋)
27		fveq2 6882	. . . 4 ⊢ ((𝑁‘𝑥) = (𝑁‘𝑦) → (𝑁‘(𝑁‘𝑥)) = (𝑁‘(𝑁‘𝑦)))
28	4, 6	grpo2inv 30823	. . . . . 6 ⊢ ((𝐺 ∈ GrpOp ∧ 𝑥 ∈ 𝑋) → (𝑁‘(𝑁‘𝑥)) = 𝑥)
29	28, 13	eqeqan12d 2783	. . . . 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 3214	. 2 ⊢ (𝐺 ∈ GrpOp → ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ((𝑁‘𝑥) = (𝑁‘𝑦) → 𝑥 = 𝑦))
33		dff1o6 7274	. 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 1360	1 ⊢ (𝐺 ∈ GrpOp → 𝑁:𝑋–1-1-onto→𝑋)

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   = wceq 1567   ∈ wcel 2149  {cab 2747  ∀wral 3085  ∃wrex 3095   ↦ cmpt 5196  ran crn 5663   Fn wfn 6532  –1-1-onto→wf1o 6536  ‘cfv 6537  ℩crio 7367  (class class class)co 7411  GrpOpcgr 30781  GIdcgi 30782  invcgn 30783

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-rep 5242  ax-sep 5261  ax-nul 5271  ax-pr 5405  ax-un 7733

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-ral 3086  df-rex 3096  df-reu 3377  df-rab 3424  df-v 3465  df-sbc 3754  df-csb 3862  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4493  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4877  df-iun 4962  df-br 5114  df-opab 5178  df-mpt 5197  df-id 5557  df-xp 5668  df-rel 5669  df-cnv 5670  df-co 5671  df-dm 5672  df-rn 5673  df-res 5674  df-ima 5675  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-riota 7368  df-ov 7414  df-grpo 30785  df-gid 30786  df-ginv 30787

This theorem is referenced by:  nvinvfval  30932