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Theorem grpoinvf 30551
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:𝑋1-1-onto𝑋)

Proof of Theorem grpoinvf
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 riotaex 7392 . . . 4 (𝑦𝑋 (𝑦𝐺𝑥) = (GId‘𝐺)) ∈ V
2 eqid 2737 . . . 4 (𝑥𝑋 ↦ (𝑦𝑋 (𝑦𝐺𝑥) = (GId‘𝐺))) = (𝑥𝑋 ↦ (𝑦𝑋 (𝑦𝐺𝑥) = (GId‘𝐺)))
31, 2fnmpti 6711 . . 3 (𝑥𝑋 ↦ (𝑦𝑋 (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)
74, 5, 6grpoinvfval 30541 . . . 4 (𝐺 ∈ GrpOp → 𝑁 = (𝑥𝑋 ↦ (𝑦𝑋 (𝑦𝐺𝑥) = (GId‘𝐺))))
87fneq1d 6661 . . 3 (𝐺 ∈ GrpOp → (𝑁 Fn 𝑋 ↔ (𝑥𝑋 ↦ (𝑦𝑋 (𝑦𝐺𝑥) = (GId‘𝐺))) Fn 𝑋))
93, 8mpbiri 258 . 2 (𝐺 ∈ GrpOp → 𝑁 Fn 𝑋)
10 fnrnfv 6968 . . . 4 (𝑁 Fn 𝑋 → ran 𝑁 = {𝑦 ∣ ∃𝑥𝑋 𝑦 = (𝑁𝑥)})
119, 10syl 17 . . 3 (𝐺 ∈ GrpOp → ran 𝑁 = {𝑦 ∣ ∃𝑥𝑋 𝑦 = (𝑁𝑥)})
124, 6grpoinvcl 30543 . . . . . . 7 ((𝐺 ∈ GrpOp ∧ 𝑦𝑋) → (𝑁𝑦) ∈ 𝑋)
134, 6grpo2inv 30550 . . . . . . . 8 ((𝐺 ∈ GrpOp ∧ 𝑦𝑋) → (𝑁‘(𝑁𝑦)) = 𝑦)
1413eqcomd 2743 . . . . . . 7 ((𝐺 ∈ GrpOp ∧ 𝑦𝑋) → 𝑦 = (𝑁‘(𝑁𝑦)))
15 fveq2 6906 . . . . . . . 8 (𝑥 = (𝑁𝑦) → (𝑁𝑥) = (𝑁‘(𝑁𝑦)))
1615rspceeqv 3645 . . . . . . 7 (((𝑁𝑦) ∈ 𝑋𝑦 = (𝑁‘(𝑁𝑦))) → ∃𝑥𝑋 𝑦 = (𝑁𝑥))
1712, 14, 16syl2anc 584 . . . . . 6 ((𝐺 ∈ GrpOp ∧ 𝑦𝑋) → ∃𝑥𝑋 𝑦 = (𝑁𝑥))
1817ex 412 . . . . 5 (𝐺 ∈ GrpOp → (𝑦𝑋 → ∃𝑥𝑋 𝑦 = (𝑁𝑥)))
19 simpr 484 . . . . . . 7 (((𝐺 ∈ GrpOp ∧ 𝑥𝑋) ∧ 𝑦 = (𝑁𝑥)) → 𝑦 = (𝑁𝑥))
204, 6grpoinvcl 30543 . . . . . . . 8 ((𝐺 ∈ GrpOp ∧ 𝑥𝑋) → (𝑁𝑥) ∈ 𝑋)
2120adantr 480 . . . . . . 7 (((𝐺 ∈ GrpOp ∧ 𝑥𝑋) ∧ 𝑦 = (𝑁𝑥)) → (𝑁𝑥) ∈ 𝑋)
2219, 21eqeltrd 2841 . . . . . 6 (((𝐺 ∈ GrpOp ∧ 𝑥𝑋) ∧ 𝑦 = (𝑁𝑥)) → 𝑦𝑋)
2322rexlimdva2 3157 . . . . 5 (𝐺 ∈ GrpOp → (∃𝑥𝑋 𝑦 = (𝑁𝑥) → 𝑦𝑋))
2418, 23impbid 212 . . . 4 (𝐺 ∈ GrpOp → (𝑦𝑋 ↔ ∃𝑥𝑋 𝑦 = (𝑁𝑥)))
2524eqabdv 2875 . . 3 (𝐺 ∈ GrpOp → 𝑋 = {𝑦 ∣ ∃𝑥𝑋 𝑦 = (𝑁𝑥)})
2611, 25eqtr4d 2780 . 2 (𝐺 ∈ GrpOp → ran 𝑁 = 𝑋)
27 fveq2 6906 . . . 4 ((𝑁𝑥) = (𝑁𝑦) → (𝑁‘(𝑁𝑥)) = (𝑁‘(𝑁𝑦)))
284, 6grpo2inv 30550 . . . . . 6 ((𝐺 ∈ GrpOp ∧ 𝑥𝑋) → (𝑁‘(𝑁𝑥)) = 𝑥)
2928, 13eqeqan12d 2751 . . . . 5 (((𝐺 ∈ GrpOp ∧ 𝑥𝑋) ∧ (𝐺 ∈ GrpOp ∧ 𝑦𝑋)) → ((𝑁‘(𝑁𝑥)) = (𝑁‘(𝑁𝑦)) ↔ 𝑥 = 𝑦))
3029anandis 678 . . . 4 ((𝐺 ∈ GrpOp ∧ (𝑥𝑋𝑦𝑋)) → ((𝑁‘(𝑁𝑥)) = (𝑁‘(𝑁𝑦)) ↔ 𝑥 = 𝑦))
3127, 30imbitrid 244 . . 3 ((𝐺 ∈ GrpOp ∧ (𝑥𝑋𝑦𝑋)) → ((𝑁𝑥) = (𝑁𝑦) → 𝑥 = 𝑦))
3231ralrimivva 3202 . 2 (𝐺 ∈ GrpOp → ∀𝑥𝑋𝑦𝑋 ((𝑁𝑥) = (𝑁𝑦) → 𝑥 = 𝑦))
33 dff1o6 7295 . 2 (𝑁:𝑋1-1-onto𝑋 ↔ (𝑁 Fn 𝑋 ∧ ran 𝑁 = 𝑋 ∧ ∀𝑥𝑋𝑦𝑋 ((𝑁𝑥) = (𝑁𝑦) → 𝑥 = 𝑦)))
349, 26, 32, 33syl3anbrc 1344 1 (𝐺 ∈ GrpOp → 𝑁:𝑋1-1-onto𝑋)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1540  wcel 2108  {cab 2714  wral 3061  wrex 3070  cmpt 5225  ran crn 5686   Fn wfn 6556  1-1-ontowf1o 6560  cfv 6561  crio 7387  (class class class)co 7431  GrpOpcgr 30508  GIdcgi 30509  invcgn 30510
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 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7388  df-ov 7434  df-grpo 30512  df-gid 30513  df-ginv 30514
This theorem is referenced by:  nvinvfval  30659
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