Theorem grpoinvf 30468

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 7351	. . . 4 ⊢ (℩𝑦 ∈ 𝑋 (𝑦𝐺𝑥) = (GId‘𝐺)) ∈ V
2		eqid 2730	. . . 4 ⊢ (𝑥 ∈ 𝑋 ↦ (℩𝑦 ∈ 𝑋 (𝑦𝐺𝑥) = (GId‘𝐺))) = (𝑥 ∈ 𝑋 ↦ (℩𝑦 ∈ 𝑋 (𝑦𝐺𝑥) = (GId‘𝐺)))
3	1, 2	fnmpti 6664	. . 3 ⊢ (𝑥 ∈ 𝑋 ↦ (℩𝑦 ∈ 𝑋 (𝑦𝐺𝑥) = (GId‘𝐺))) Fn 𝑋
4		grpasscan1.1	. . . . 5 ⊢ 𝑋 = ran 𝐺
5		eqid 2730	. . . . 5 ⊢ (GId‘𝐺) = (GId‘𝐺)
6		grpasscan1.2	. . . . 5 ⊢ 𝑁 = (inv‘𝐺)
7	4, 5, 6	grpoinvfval 30458	. . . 4 ⊢ (𝐺 ∈ GrpOp → 𝑁 = (𝑥 ∈ 𝑋 ↦ (℩𝑦 ∈ 𝑋 (𝑦𝐺𝑥) = (GId‘𝐺))))
8	7	fneq1d 6614	. . 3 ⊢ (𝐺 ∈ GrpOp → (𝑁 Fn 𝑋 ↔ (𝑥 ∈ 𝑋 ↦ (℩𝑦 ∈ 𝑋 (𝑦𝐺𝑥) = (GId‘𝐺))) Fn 𝑋))
9	3, 8	mpbiri 258	. 2 ⊢ (𝐺 ∈ GrpOp → 𝑁 Fn 𝑋)
10		fnrnfv 6923	. . . 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 30460	. . . . . . 7 ⊢ ((𝐺 ∈ GrpOp ∧ 𝑦 ∈ 𝑋) → (𝑁‘𝑦) ∈ 𝑋)
13	4, 6	grpo2inv 30467	. . . . . . . 8 ⊢ ((𝐺 ∈ GrpOp ∧ 𝑦 ∈ 𝑋) → (𝑁‘(𝑁‘𝑦)) = 𝑦)
14	13	eqcomd 2736	. . . . . . 7 ⊢ ((𝐺 ∈ GrpOp ∧ 𝑦 ∈ 𝑋) → 𝑦 = (𝑁‘(𝑁‘𝑦)))
15		fveq2 6861	. . . . . . . 8 ⊢ (𝑥 = (𝑁‘𝑦) → (𝑁‘𝑥) = (𝑁‘(𝑁‘𝑦)))
16	15	rspceeqv 3614	. . . . . . 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 412	. . . . 5 ⊢ (𝐺 ∈ GrpOp → (𝑦 ∈ 𝑋 → ∃𝑥 ∈ 𝑋 𝑦 = (𝑁‘𝑥)))
19		simpr 484	. . . . . . 7 ⊢ (((𝐺 ∈ GrpOp ∧ 𝑥 ∈ 𝑋) ∧ 𝑦 = (𝑁‘𝑥)) → 𝑦 = (𝑁‘𝑥))
20	4, 6	grpoinvcl 30460	. . . . . . . 8 ⊢ ((𝐺 ∈ GrpOp ∧ 𝑥 ∈ 𝑋) → (𝑁‘𝑥) ∈ 𝑋)
21	20	adantr 480	. . . . . . 7 ⊢ (((𝐺 ∈ GrpOp ∧ 𝑥 ∈ 𝑋) ∧ 𝑦 = (𝑁‘𝑥)) → (𝑁‘𝑥) ∈ 𝑋)
22	19, 21	eqeltrd 2829	. . . . . 6 ⊢ (((𝐺 ∈ GrpOp ∧ 𝑥 ∈ 𝑋) ∧ 𝑦 = (𝑁‘𝑥)) → 𝑦 ∈ 𝑋)
23	22	rexlimdva2 3137	. . . . 5 ⊢ (𝐺 ∈ GrpOp → (∃𝑥 ∈ 𝑋 𝑦 = (𝑁‘𝑥) → 𝑦 ∈ 𝑋))
24	18, 23	impbid 212	. . . 4 ⊢ (𝐺 ∈ GrpOp → (𝑦 ∈ 𝑋 ↔ ∃𝑥 ∈ 𝑋 𝑦 = (𝑁‘𝑥)))
25	24	eqabdv 2862	. . 3 ⊢ (𝐺 ∈ GrpOp → 𝑋 = {𝑦 ∣ ∃𝑥 ∈ 𝑋 𝑦 = (𝑁‘𝑥)})
26	11, 25	eqtr4d 2768	. 2 ⊢ (𝐺 ∈ GrpOp → ran 𝑁 = 𝑋)
27		fveq2 6861	. . . 4 ⊢ ((𝑁‘𝑥) = (𝑁‘𝑦) → (𝑁‘(𝑁‘𝑥)) = (𝑁‘(𝑁‘𝑦)))
28	4, 6	grpo2inv 30467	. . . . . 6 ⊢ ((𝐺 ∈ GrpOp ∧ 𝑥 ∈ 𝑋) → (𝑁‘(𝑁‘𝑥)) = 𝑥)
29	28, 13	eqeqan12d 2744	. . . . 5 ⊢ (((𝐺 ∈ GrpOp ∧ 𝑥 ∈ 𝑋) ∧ (𝐺 ∈ GrpOp ∧ 𝑦 ∈ 𝑋)) → ((𝑁‘(𝑁‘𝑥)) = (𝑁‘(𝑁‘𝑦)) ↔ 𝑥 = 𝑦))
30	29	anandis 678	. . . 4 ⊢ ((𝐺 ∈ GrpOp ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → ((𝑁‘(𝑁‘𝑥)) = (𝑁‘(𝑁‘𝑦)) ↔ 𝑥 = 𝑦))
31	27, 30	imbitrid 244	. . 3 ⊢ ((𝐺 ∈ GrpOp ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → ((𝑁‘𝑥) = (𝑁‘𝑦) → 𝑥 = 𝑦))
32	31	ralrimivva 3181	. 2 ⊢ (𝐺 ∈ GrpOp → ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ((𝑁‘𝑥) = (𝑁‘𝑦) → 𝑥 = 𝑦))
33		dff1o6 7253	. 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 1344	1 ⊢ (𝐺 ∈ GrpOp → 𝑁:𝑋–1-1-onto→𝑋)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2109  {cab 2708  ∀wral 3045  ∃wrex 3054   ↦ cmpt 5191  ran crn 5642   Fn wfn 6509  –1-1-onto→wf1o 6513  ‘cfv 6514  ℩crio 7346  (class class class)co 7390  GrpOpcgr 30425  GIdcgi 30426  invcgn 30427

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-rep 5237  ax-sep 5254  ax-nul 5264  ax-pr 5390  ax-un 7714

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-reu 3357  df-rab 3409  df-v 3452  df-sbc 3757  df-csb 3866  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-nul 4300  df-if 4492  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-iun 4960  df-br 5111  df-opab 5173  df-mpt 5192  df-id 5536  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-res 5653  df-ima 5654  df-iota 6467  df-fun 6516  df-fn 6517  df-f 6518  df-f1 6519  df-fo 6520  df-f1o 6521  df-fv 6522  df-riota 7347  df-ov 7393  df-grpo 30429  df-gid 30430  df-ginv 30431

This theorem is referenced by:  nvinvfval  30576