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Theorem grpoinvf 27616
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 6730 . . . 4 (𝑦𝑋 (𝑦𝐺𝑥) = (GId‘𝐺)) ∈ V
2 eqid 2724 . . . 4 (𝑥𝑋 ↦ (𝑦𝑋 (𝑦𝐺𝑥) = (GId‘𝐺))) = (𝑥𝑋 ↦ (𝑦𝑋 (𝑦𝐺𝑥) = (GId‘𝐺)))
31, 2fnmpti 6135 . . 3 (𝑥𝑋 ↦ (𝑦𝑋 (𝑦𝐺𝑥) = (GId‘𝐺))) Fn 𝑋
4 grpasscan1.1 . . . . 5 𝑋 = ran 𝐺
5 eqid 2724 . . . . 5 (GId‘𝐺) = (GId‘𝐺)
6 grpasscan1.2 . . . . 5 𝑁 = (inv‘𝐺)
74, 5, 6grpoinvfval 27606 . . . 4 (𝐺 ∈ GrpOp → 𝑁 = (𝑥𝑋 ↦ (𝑦𝑋 (𝑦𝐺𝑥) = (GId‘𝐺))))
87fneq1d 6094 . . 3 (𝐺 ∈ GrpOp → (𝑁 Fn 𝑋 ↔ (𝑥𝑋 ↦ (𝑦𝑋 (𝑦𝐺𝑥) = (GId‘𝐺))) Fn 𝑋))
93, 8mpbiri 248 . 2 (𝐺 ∈ GrpOp → 𝑁 Fn 𝑋)
10 fnrnfv 6356 . . . 4 (𝑁 Fn 𝑋 → ran 𝑁 = {𝑦 ∣ ∃𝑥𝑋 𝑦 = (𝑁𝑥)})
119, 10syl 17 . . 3 (𝐺 ∈ GrpOp → ran 𝑁 = {𝑦 ∣ ∃𝑥𝑋 𝑦 = (𝑁𝑥)})
124, 6grpoinvcl 27608 . . . . . . 7 ((𝐺 ∈ GrpOp ∧ 𝑦𝑋) → (𝑁𝑦) ∈ 𝑋)
134, 6grpo2inv 27615 . . . . . . . 8 ((𝐺 ∈ GrpOp ∧ 𝑦𝑋) → (𝑁‘(𝑁𝑦)) = 𝑦)
1413eqcomd 2730 . . . . . . 7 ((𝐺 ∈ GrpOp ∧ 𝑦𝑋) → 𝑦 = (𝑁‘(𝑁𝑦)))
15 fveq2 6304 . . . . . . . . 9 (𝑥 = (𝑁𝑦) → (𝑁𝑥) = (𝑁‘(𝑁𝑦)))
1615eqeq2d 2734 . . . . . . . 8 (𝑥 = (𝑁𝑦) → (𝑦 = (𝑁𝑥) ↔ 𝑦 = (𝑁‘(𝑁𝑦))))
1716rspcev 3413 . . . . . . 7 (((𝑁𝑦) ∈ 𝑋𝑦 = (𝑁‘(𝑁𝑦))) → ∃𝑥𝑋 𝑦 = (𝑁𝑥))
1812, 14, 17syl2anc 696 . . . . . 6 ((𝐺 ∈ GrpOp ∧ 𝑦𝑋) → ∃𝑥𝑋 𝑦 = (𝑁𝑥))
1918ex 449 . . . . 5 (𝐺 ∈ GrpOp → (𝑦𝑋 → ∃𝑥𝑋 𝑦 = (𝑁𝑥)))
20 simpr 479 . . . . . . . 8 (((𝐺 ∈ GrpOp ∧ 𝑥𝑋) ∧ 𝑦 = (𝑁𝑥)) → 𝑦 = (𝑁𝑥))
214, 6grpoinvcl 27608 . . . . . . . . 9 ((𝐺 ∈ GrpOp ∧ 𝑥𝑋) → (𝑁𝑥) ∈ 𝑋)
2221adantr 472 . . . . . . . 8 (((𝐺 ∈ GrpOp ∧ 𝑥𝑋) ∧ 𝑦 = (𝑁𝑥)) → (𝑁𝑥) ∈ 𝑋)
2320, 22eqeltrd 2803 . . . . . . 7 (((𝐺 ∈ GrpOp ∧ 𝑥𝑋) ∧ 𝑦 = (𝑁𝑥)) → 𝑦𝑋)
2423exp31 631 . . . . . 6 (𝐺 ∈ GrpOp → (𝑥𝑋 → (𝑦 = (𝑁𝑥) → 𝑦𝑋)))
2524rexlimdv 3132 . . . . 5 (𝐺 ∈ GrpOp → (∃𝑥𝑋 𝑦 = (𝑁𝑥) → 𝑦𝑋))
2619, 25impbid 202 . . . 4 (𝐺 ∈ GrpOp → (𝑦𝑋 ↔ ∃𝑥𝑋 𝑦 = (𝑁𝑥)))
2726abbi2dv 2844 . . 3 (𝐺 ∈ GrpOp → 𝑋 = {𝑦 ∣ ∃𝑥𝑋 𝑦 = (𝑁𝑥)})
2811, 27eqtr4d 2761 . 2 (𝐺 ∈ GrpOp → ran 𝑁 = 𝑋)
29 fveq2 6304 . . . 4 ((𝑁𝑥) = (𝑁𝑦) → (𝑁‘(𝑁𝑥)) = (𝑁‘(𝑁𝑦)))
304, 6grpo2inv 27615 . . . . . 6 ((𝐺 ∈ GrpOp ∧ 𝑥𝑋) → (𝑁‘(𝑁𝑥)) = 𝑥)
3130, 13eqeqan12d 2740 . . . . 5 (((𝐺 ∈ GrpOp ∧ 𝑥𝑋) ∧ (𝐺 ∈ GrpOp ∧ 𝑦𝑋)) → ((𝑁‘(𝑁𝑥)) = (𝑁‘(𝑁𝑦)) ↔ 𝑥 = 𝑦))
3231anandis 908 . . . 4 ((𝐺 ∈ GrpOp ∧ (𝑥𝑋𝑦𝑋)) → ((𝑁‘(𝑁𝑥)) = (𝑁‘(𝑁𝑦)) ↔ 𝑥 = 𝑦))
3329, 32syl5ib 234 . . 3 ((𝐺 ∈ GrpOp ∧ (𝑥𝑋𝑦𝑋)) → ((𝑁𝑥) = (𝑁𝑦) → 𝑥 = 𝑦))
3433ralrimivva 3073 . 2 (𝐺 ∈ GrpOp → ∀𝑥𝑋𝑦𝑋 ((𝑁𝑥) = (𝑁𝑦) → 𝑥 = 𝑦))
35 dff1o6 6646 . 2 (𝑁:𝑋1-1-onto𝑋 ↔ (𝑁 Fn 𝑋 ∧ ran 𝑁 = 𝑋 ∧ ∀𝑥𝑋𝑦𝑋 ((𝑁𝑥) = (𝑁𝑦) → 𝑥 = 𝑦)))
369, 28, 34, 35syl3anbrc 1383 1 (𝐺 ∈ GrpOp → 𝑁:𝑋1-1-onto𝑋)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383   = wceq 1596  wcel 2103  {cab 2710  wral 3014  wrex 3015  cmpt 4837  ran crn 5219   Fn wfn 5996  1-1-ontowf1o 6000  cfv 6001  crio 6725  (class class class)co 6765  GrpOpcgr 27573  GIdcgi 27574  invcgn 27575
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1835  ax-4 1850  ax-5 1952  ax-6 2018  ax-7 2054  ax-8 2105  ax-9 2112  ax-10 2132  ax-11 2147  ax-12 2160  ax-13 2355  ax-ext 2704  ax-rep 4879  ax-sep 4889  ax-nul 4897  ax-pr 5011  ax-un 7066
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1599  df-ex 1818  df-nf 1823  df-sb 2011  df-eu 2575  df-mo 2576  df-clab 2711  df-cleq 2717  df-clel 2720  df-nfc 2855  df-ne 2897  df-ral 3019  df-rex 3020  df-reu 3021  df-rab 3023  df-v 3306  df-sbc 3542  df-csb 3640  df-dif 3683  df-un 3685  df-in 3687  df-ss 3694  df-nul 4024  df-if 4195  df-sn 4286  df-pr 4288  df-op 4292  df-uni 4545  df-iun 4630  df-br 4761  df-opab 4821  df-mpt 4838  df-id 5128  df-xp 5224  df-rel 5225  df-cnv 5226  df-co 5227  df-dm 5228  df-rn 5229  df-res 5230  df-ima 5231  df-iota 5964  df-fun 6003  df-fn 6004  df-f 6005  df-f1 6006  df-fo 6007  df-f1o 6008  df-fv 6009  df-riota 6726  df-ov 6768  df-grpo 27577  df-gid 27578  df-ginv 27579
This theorem is referenced by:  nvinvfval  27725
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