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Theorem grpoinvf 28318
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 7111 . . . 4 (𝑦𝑋 (𝑦𝐺𝑥) = (GId‘𝐺)) ∈ V
2 eqid 2824 . . . 4 (𝑥𝑋 ↦ (𝑦𝑋 (𝑦𝐺𝑥) = (GId‘𝐺))) = (𝑥𝑋 ↦ (𝑦𝑋 (𝑦𝐺𝑥) = (GId‘𝐺)))
31, 2fnmpti 6480 . . 3 (𝑥𝑋 ↦ (𝑦𝑋 (𝑦𝐺𝑥) = (GId‘𝐺))) Fn 𝑋
4 grpasscan1.1 . . . . 5 𝑋 = ran 𝐺
5 eqid 2824 . . . . 5 (GId‘𝐺) = (GId‘𝐺)
6 grpasscan1.2 . . . . 5 𝑁 = (inv‘𝐺)
74, 5, 6grpoinvfval 28308 . . . 4 (𝐺 ∈ GrpOp → 𝑁 = (𝑥𝑋 ↦ (𝑦𝑋 (𝑦𝐺𝑥) = (GId‘𝐺))))
87fneq1d 6434 . . 3 (𝐺 ∈ GrpOp → (𝑁 Fn 𝑋 ↔ (𝑥𝑋 ↦ (𝑦𝑋 (𝑦𝐺𝑥) = (GId‘𝐺))) Fn 𝑋))
93, 8mpbiri 261 . 2 (𝐺 ∈ GrpOp → 𝑁 Fn 𝑋)
10 fnrnfv 6716 . . . 4 (𝑁 Fn 𝑋 → ran 𝑁 = {𝑦 ∣ ∃𝑥𝑋 𝑦 = (𝑁𝑥)})
119, 10syl 17 . . 3 (𝐺 ∈ GrpOp → ran 𝑁 = {𝑦 ∣ ∃𝑥𝑋 𝑦 = (𝑁𝑥)})
124, 6grpoinvcl 28310 . . . . . . 7 ((𝐺 ∈ GrpOp ∧ 𝑦𝑋) → (𝑁𝑦) ∈ 𝑋)
134, 6grpo2inv 28317 . . . . . . . 8 ((𝐺 ∈ GrpOp ∧ 𝑦𝑋) → (𝑁‘(𝑁𝑦)) = 𝑦)
1413eqcomd 2830 . . . . . . 7 ((𝐺 ∈ GrpOp ∧ 𝑦𝑋) → 𝑦 = (𝑁‘(𝑁𝑦)))
15 fveq2 6661 . . . . . . . 8 (𝑥 = (𝑁𝑦) → (𝑁𝑥) = (𝑁‘(𝑁𝑦)))
1615rspceeqv 3624 . . . . . . 7 (((𝑁𝑦) ∈ 𝑋𝑦 = (𝑁‘(𝑁𝑦))) → ∃𝑥𝑋 𝑦 = (𝑁𝑥))
1712, 14, 16syl2anc 587 . . . . . 6 ((𝐺 ∈ GrpOp ∧ 𝑦𝑋) → ∃𝑥𝑋 𝑦 = (𝑁𝑥))
1817ex 416 . . . . 5 (𝐺 ∈ GrpOp → (𝑦𝑋 → ∃𝑥𝑋 𝑦 = (𝑁𝑥)))
19 simpr 488 . . . . . . 7 (((𝐺 ∈ GrpOp ∧ 𝑥𝑋) ∧ 𝑦 = (𝑁𝑥)) → 𝑦 = (𝑁𝑥))
204, 6grpoinvcl 28310 . . . . . . . 8 ((𝐺 ∈ GrpOp ∧ 𝑥𝑋) → (𝑁𝑥) ∈ 𝑋)
2120adantr 484 . . . . . . 7 (((𝐺 ∈ GrpOp ∧ 𝑥𝑋) ∧ 𝑦 = (𝑁𝑥)) → (𝑁𝑥) ∈ 𝑋)
2219, 21eqeltrd 2916 . . . . . 6 (((𝐺 ∈ GrpOp ∧ 𝑥𝑋) ∧ 𝑦 = (𝑁𝑥)) → 𝑦𝑋)
2322rexlimdva2 3279 . . . . 5 (𝐺 ∈ GrpOp → (∃𝑥𝑋 𝑦 = (𝑁𝑥) → 𝑦𝑋))
2418, 23impbid 215 . . . 4 (𝐺 ∈ GrpOp → (𝑦𝑋 ↔ ∃𝑥𝑋 𝑦 = (𝑁𝑥)))
2524abbi2dv 2953 . . 3 (𝐺 ∈ GrpOp → 𝑋 = {𝑦 ∣ ∃𝑥𝑋 𝑦 = (𝑁𝑥)})
2611, 25eqtr4d 2862 . 2 (𝐺 ∈ GrpOp → ran 𝑁 = 𝑋)
27 fveq2 6661 . . . 4 ((𝑁𝑥) = (𝑁𝑦) → (𝑁‘(𝑁𝑥)) = (𝑁‘(𝑁𝑦)))
284, 6grpo2inv 28317 . . . . . 6 ((𝐺 ∈ GrpOp ∧ 𝑥𝑋) → (𝑁‘(𝑁𝑥)) = 𝑥)
2928, 13eqeqan12d 2841 . . . . 5 (((𝐺 ∈ GrpOp ∧ 𝑥𝑋) ∧ (𝐺 ∈ GrpOp ∧ 𝑦𝑋)) → ((𝑁‘(𝑁𝑥)) = (𝑁‘(𝑁𝑦)) ↔ 𝑥 = 𝑦))
3029anandis 677 . . . 4 ((𝐺 ∈ GrpOp ∧ (𝑥𝑋𝑦𝑋)) → ((𝑁‘(𝑁𝑥)) = (𝑁‘(𝑁𝑦)) ↔ 𝑥 = 𝑦))
3127, 30syl5ib 247 . . 3 ((𝐺 ∈ GrpOp ∧ (𝑥𝑋𝑦𝑋)) → ((𝑁𝑥) = (𝑁𝑦) → 𝑥 = 𝑦))
3231ralrimivva 3186 . 2 (𝐺 ∈ GrpOp → ∀𝑥𝑋𝑦𝑋 ((𝑁𝑥) = (𝑁𝑦) → 𝑥 = 𝑦))
33 dff1o6 7024 . 2 (𝑁:𝑋1-1-onto𝑋 ↔ (𝑁 Fn 𝑋 ∧ ran 𝑁 = 𝑋 ∧ ∀𝑥𝑋𝑦𝑋 ((𝑁𝑥) = (𝑁𝑦) → 𝑥 = 𝑦)))
349, 26, 32, 33syl3anbrc 1340 1 (𝐺 ∈ GrpOp → 𝑁:𝑋1-1-onto𝑋)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399   = wceq 1538  wcel 2115  {cab 2802  wral 3133  wrex 3134  cmpt 5132  ran crn 5543   Fn wfn 6338  1-1-ontowf1o 6342  cfv 6343  crio 7106  (class class class)co 7149  GrpOpcgr 28275  GIdcgi 28276  invcgn 28277
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-rep 5176  ax-sep 5189  ax-nul 5196  ax-pr 5317  ax-un 7455
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-ral 3138  df-rex 3139  df-reu 3140  df-rab 3142  df-v 3482  df-sbc 3759  df-csb 3867  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-if 4451  df-sn 4551  df-pr 4553  df-op 4557  df-uni 4825  df-iun 4907  df-br 5053  df-opab 5115  df-mpt 5133  df-id 5447  df-xp 5548  df-rel 5549  df-cnv 5550  df-co 5551  df-dm 5552  df-rn 5553  df-res 5554  df-ima 5555  df-iota 6302  df-fun 6345  df-fn 6346  df-f 6347  df-f1 6348  df-fo 6349  df-f1o 6350  df-fv 6351  df-riota 7107  df-ov 7152  df-grpo 28279  df-gid 28280  df-ginv 28281
This theorem is referenced by:  nvinvfval  28426
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