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Theorem grpoinvf 27232
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 6569 . . . 4 (𝑦𝑋 (𝑦𝐺𝑥) = (GId‘𝐺)) ∈ V
2 eqid 2621 . . . 4 (𝑥𝑋 ↦ (𝑦𝑋 (𝑦𝐺𝑥) = (GId‘𝐺))) = (𝑥𝑋 ↦ (𝑦𝑋 (𝑦𝐺𝑥) = (GId‘𝐺)))
31, 2fnmpti 5979 . . 3 (𝑥𝑋 ↦ (𝑦𝑋 (𝑦𝐺𝑥) = (GId‘𝐺))) Fn 𝑋
4 grpasscan1.1 . . . . 5 𝑋 = ran 𝐺
5 eqid 2621 . . . . 5 (GId‘𝐺) = (GId‘𝐺)
6 grpasscan1.2 . . . . 5 𝑁 = (inv‘𝐺)
74, 5, 6grpoinvfval 27222 . . . 4 (𝐺 ∈ GrpOp → 𝑁 = (𝑥𝑋 ↦ (𝑦𝑋 (𝑦𝐺𝑥) = (GId‘𝐺))))
87fneq1d 5939 . . 3 (𝐺 ∈ GrpOp → (𝑁 Fn 𝑋 ↔ (𝑥𝑋 ↦ (𝑦𝑋 (𝑦𝐺𝑥) = (GId‘𝐺))) Fn 𝑋))
93, 8mpbiri 248 . 2 (𝐺 ∈ GrpOp → 𝑁 Fn 𝑋)
10 fnrnfv 6199 . . . 4 (𝑁 Fn 𝑋 → ran 𝑁 = {𝑦 ∣ ∃𝑥𝑋 𝑦 = (𝑁𝑥)})
119, 10syl 17 . . 3 (𝐺 ∈ GrpOp → ran 𝑁 = {𝑦 ∣ ∃𝑥𝑋 𝑦 = (𝑁𝑥)})
124, 6grpoinvcl 27224 . . . . . . 7 ((𝐺 ∈ GrpOp ∧ 𝑦𝑋) → (𝑁𝑦) ∈ 𝑋)
134, 6grpo2inv 27231 . . . . . . . 8 ((𝐺 ∈ GrpOp ∧ 𝑦𝑋) → (𝑁‘(𝑁𝑦)) = 𝑦)
1413eqcomd 2627 . . . . . . 7 ((𝐺 ∈ GrpOp ∧ 𝑦𝑋) → 𝑦 = (𝑁‘(𝑁𝑦)))
15 fveq2 6148 . . . . . . . . 9 (𝑥 = (𝑁𝑦) → (𝑁𝑥) = (𝑁‘(𝑁𝑦)))
1615eqeq2d 2631 . . . . . . . 8 (𝑥 = (𝑁𝑦) → (𝑦 = (𝑁𝑥) ↔ 𝑦 = (𝑁‘(𝑁𝑦))))
1716rspcev 3295 . . . . . . 7 (((𝑁𝑦) ∈ 𝑋𝑦 = (𝑁‘(𝑁𝑦))) → ∃𝑥𝑋 𝑦 = (𝑁𝑥))
1812, 14, 17syl2anc 692 . . . . . 6 ((𝐺 ∈ GrpOp ∧ 𝑦𝑋) → ∃𝑥𝑋 𝑦 = (𝑁𝑥))
1918ex 450 . . . . 5 (𝐺 ∈ GrpOp → (𝑦𝑋 → ∃𝑥𝑋 𝑦 = (𝑁𝑥)))
20 simpr 477 . . . . . . . 8 (((𝐺 ∈ GrpOp ∧ 𝑥𝑋) ∧ 𝑦 = (𝑁𝑥)) → 𝑦 = (𝑁𝑥))
214, 6grpoinvcl 27224 . . . . . . . . 9 ((𝐺 ∈ GrpOp ∧ 𝑥𝑋) → (𝑁𝑥) ∈ 𝑋)
2221adantr 481 . . . . . . . 8 (((𝐺 ∈ GrpOp ∧ 𝑥𝑋) ∧ 𝑦 = (𝑁𝑥)) → (𝑁𝑥) ∈ 𝑋)
2320, 22eqeltrd 2698 . . . . . . 7 (((𝐺 ∈ GrpOp ∧ 𝑥𝑋) ∧ 𝑦 = (𝑁𝑥)) → 𝑦𝑋)
2423exp31 629 . . . . . 6 (𝐺 ∈ GrpOp → (𝑥𝑋 → (𝑦 = (𝑁𝑥) → 𝑦𝑋)))
2524rexlimdv 3023 . . . . 5 (𝐺 ∈ GrpOp → (∃𝑥𝑋 𝑦 = (𝑁𝑥) → 𝑦𝑋))
2619, 25impbid 202 . . . 4 (𝐺 ∈ GrpOp → (𝑦𝑋 ↔ ∃𝑥𝑋 𝑦 = (𝑁𝑥)))
2726abbi2dv 2739 . . 3 (𝐺 ∈ GrpOp → 𝑋 = {𝑦 ∣ ∃𝑥𝑋 𝑦 = (𝑁𝑥)})
2811, 27eqtr4d 2658 . 2 (𝐺 ∈ GrpOp → ran 𝑁 = 𝑋)
29 fveq2 6148 . . . 4 ((𝑁𝑥) = (𝑁𝑦) → (𝑁‘(𝑁𝑥)) = (𝑁‘(𝑁𝑦)))
304, 6grpo2inv 27231 . . . . . 6 ((𝐺 ∈ GrpOp ∧ 𝑥𝑋) → (𝑁‘(𝑁𝑥)) = 𝑥)
3130, 13eqeqan12d 2637 . . . . 5 (((𝐺 ∈ GrpOp ∧ 𝑥𝑋) ∧ (𝐺 ∈ GrpOp ∧ 𝑦𝑋)) → ((𝑁‘(𝑁𝑥)) = (𝑁‘(𝑁𝑦)) ↔ 𝑥 = 𝑦))
3231anandis 872 . . . 4 ((𝐺 ∈ GrpOp ∧ (𝑥𝑋𝑦𝑋)) → ((𝑁‘(𝑁𝑥)) = (𝑁‘(𝑁𝑦)) ↔ 𝑥 = 𝑦))
3329, 32syl5ib 234 . . 3 ((𝐺 ∈ GrpOp ∧ (𝑥𝑋𝑦𝑋)) → ((𝑁𝑥) = (𝑁𝑦) → 𝑥 = 𝑦))
3433ralrimivva 2965 . 2 (𝐺 ∈ GrpOp → ∀𝑥𝑋𝑦𝑋 ((𝑁𝑥) = (𝑁𝑦) → 𝑥 = 𝑦))
35 dff1o6 6485 . 2 (𝑁:𝑋1-1-onto𝑋 ↔ (𝑁 Fn 𝑋 ∧ ran 𝑁 = 𝑋 ∧ ∀𝑥𝑋𝑦𝑋 ((𝑁𝑥) = (𝑁𝑦) → 𝑥 = 𝑦)))
369, 28, 34, 35syl3anbrc 1244 1 (𝐺 ∈ GrpOp → 𝑁:𝑋1-1-onto𝑋)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384   = wceq 1480  wcel 1987  {cab 2607  wral 2907  wrex 2908  cmpt 4673  ran crn 5075   Fn wfn 5842  1-1-ontowf1o 5846  cfv 5847  crio 6564  (class class class)co 6604  GrpOpcgr 27189  GIdcgi 27190  invcgn 27191
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4731  ax-sep 4741  ax-nul 4749  ax-pr 4867  ax-un 6902
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-riota 6565  df-ov 6607  df-grpo 27193  df-gid 27194  df-ginv 27195
This theorem is referenced by:  nvinvfval  27341
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