Theorem grpoinvf 30533

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 7316	. . . 4 ⊢ (℩𝑦 ∈ 𝑋 (𝑦𝐺𝑥) = (GId‘𝐺)) ∈ V
2		eqid 2733	. . . 4 ⊢ (𝑥 ∈ 𝑋 ↦ (℩𝑦 ∈ 𝑋 (𝑦𝐺𝑥) = (GId‘𝐺))) = (𝑥 ∈ 𝑋 ↦ (℩𝑦 ∈ 𝑋 (𝑦𝐺𝑥) = (GId‘𝐺)))
3	1, 2	fnmpti 6632	. . 3 ⊢ (𝑥 ∈ 𝑋 ↦ (℩𝑦 ∈ 𝑋 (𝑦𝐺𝑥) = (GId‘𝐺))) Fn 𝑋
4		grpasscan1.1	. . . . 5 ⊢ 𝑋 = ran 𝐺
5		eqid 2733	. . . . 5 ⊢ (GId‘𝐺) = (GId‘𝐺)
6		grpasscan1.2	. . . . 5 ⊢ 𝑁 = (inv‘𝐺)
7	4, 5, 6	grpoinvfval 30523	. . . 4 ⊢ (𝐺 ∈ GrpOp → 𝑁 = (𝑥 ∈ 𝑋 ↦ (℩𝑦 ∈ 𝑋 (𝑦𝐺𝑥) = (GId‘𝐺))))
8	7	fneq1d 6582	. . 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 6890	. . . 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 30525	. . . . . . 7 ⊢ ((𝐺 ∈ GrpOp ∧ 𝑦 ∈ 𝑋) → (𝑁‘𝑦) ∈ 𝑋)
13	4, 6	grpo2inv 30532	. . . . . . . 8 ⊢ ((𝐺 ∈ GrpOp ∧ 𝑦 ∈ 𝑋) → (𝑁‘(𝑁‘𝑦)) = 𝑦)
14	13	eqcomd 2739	. . . . . . 7 ⊢ ((𝐺 ∈ GrpOp ∧ 𝑦 ∈ 𝑋) → 𝑦 = (𝑁‘(𝑁‘𝑦)))
15		fveq2 6831	. . . . . . . 8 ⊢ (𝑥 = (𝑁‘𝑦) → (𝑁‘𝑥) = (𝑁‘(𝑁‘𝑦)))
16	15	rspceeqv 3596	. . . . . . 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 30525	. . . . . . . 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 2833	. . . . . 6 ⊢ (((𝐺 ∈ GrpOp ∧ 𝑥 ∈ 𝑋) ∧ 𝑦 = (𝑁‘𝑥)) → 𝑦 ∈ 𝑋)
23	22	rexlimdva2 3136	. . . . 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 2866	. . 3 ⊢ (𝐺 ∈ GrpOp → 𝑋 = {𝑦 ∣ ∃𝑥 ∈ 𝑋 𝑦 = (𝑁‘𝑥)})
26	11, 25	eqtr4d 2771	. 2 ⊢ (𝐺 ∈ GrpOp → ran 𝑁 = 𝑋)
27		fveq2 6831	. . . 4 ⊢ ((𝑁‘𝑥) = (𝑁‘𝑦) → (𝑁‘(𝑁‘𝑥)) = (𝑁‘(𝑁‘𝑦)))
28	4, 6	grpo2inv 30532	. . . . . 6 ⊢ ((𝐺 ∈ GrpOp ∧ 𝑥 ∈ 𝑋) → (𝑁‘(𝑁‘𝑥)) = 𝑥)
29	28, 13	eqeqan12d 2747	. . . . 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 3176	. 2 ⊢ (𝐺 ∈ GrpOp → ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ((𝑁‘𝑥) = (𝑁‘𝑦) → 𝑥 = 𝑦))
33		dff1o6 7218	. 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 1541   ∈ wcel 2113  {cab 2711  ∀wral 3048  ∃wrex 3057   ↦ cmpt 5176  ran crn 5622   Fn wfn 6484  –1-1-onto→wf1o 6488  ‘cfv 6489  ℩crio 7311  (class class class)co 7355  GrpOpcgr 30490  GIdcgi 30491  invcgn 30492

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 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2705  ax-rep 5221  ax-sep 5238  ax-nul 5248  ax-pr 5374  ax-un 7677

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2725  df-clel 2808  df-nfc 2882  df-ne 2930  df-ral 3049  df-rex 3058  df-reu 3348  df-rab 3397  df-v 3439  df-sbc 3738  df-csb 3847  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4283  df-if 4477  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4861  df-iun 4945  df-br 5096  df-opab 5158  df-mpt 5177  df-id 5516  df-xp 5627  df-rel 5628  df-cnv 5629  df-co 5630  df-dm 5631  df-rn 5632  df-res 5633  df-ima 5634  df-iota 6445  df-fun 6491  df-fn 6492  df-f 6493  df-f1 6494  df-fo 6495  df-f1o 6496  df-fv 6497  df-riota 7312  df-ov 7358  df-grpo 30494  df-gid 30495  df-ginv 30496

This theorem is referenced by:  nvinvfval  30641