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Theorem grpoinvf 29516
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 = (invβ€˜πΊ)
Assertion
Ref Expression
grpoinvf (𝐺 ∈ GrpOp β†’ 𝑁:𝑋–1-1-onto→𝑋)

Proof of Theorem grpoinvf
Dummy variables π‘₯ 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 riotaex 7318 . . . 4 (℩𝑦 ∈ 𝑋 (𝑦𝐺π‘₯) = (GIdβ€˜πΊ)) ∈ V
2 eqid 2733 . . . 4 (π‘₯ ∈ 𝑋 ↦ (℩𝑦 ∈ 𝑋 (𝑦𝐺π‘₯) = (GIdβ€˜πΊ))) = (π‘₯ ∈ 𝑋 ↦ (℩𝑦 ∈ 𝑋 (𝑦𝐺π‘₯) = (GIdβ€˜πΊ)))
31, 2fnmpti 6645 . . 3 (π‘₯ ∈ 𝑋 ↦ (℩𝑦 ∈ 𝑋 (𝑦𝐺π‘₯) = (GIdβ€˜πΊ))) Fn 𝑋
4 grpasscan1.1 . . . . 5 𝑋 = ran 𝐺
5 eqid 2733 . . . . 5 (GIdβ€˜πΊ) = (GIdβ€˜πΊ)
6 grpasscan1.2 . . . . 5 𝑁 = (invβ€˜πΊ)
74, 5, 6grpoinvfval 29506 . . . 4 (𝐺 ∈ GrpOp β†’ 𝑁 = (π‘₯ ∈ 𝑋 ↦ (℩𝑦 ∈ 𝑋 (𝑦𝐺π‘₯) = (GIdβ€˜πΊ))))
87fneq1d 6596 . . 3 (𝐺 ∈ GrpOp β†’ (𝑁 Fn 𝑋 ↔ (π‘₯ ∈ 𝑋 ↦ (℩𝑦 ∈ 𝑋 (𝑦𝐺π‘₯) = (GIdβ€˜πΊ))) Fn 𝑋))
93, 8mpbiri 258 . 2 (𝐺 ∈ GrpOp β†’ 𝑁 Fn 𝑋)
10 fnrnfv 6903 . . . 4 (𝑁 Fn 𝑋 β†’ ran 𝑁 = {𝑦 ∣ βˆƒπ‘₯ ∈ 𝑋 𝑦 = (π‘β€˜π‘₯)})
119, 10syl 17 . . 3 (𝐺 ∈ GrpOp β†’ ran 𝑁 = {𝑦 ∣ βˆƒπ‘₯ ∈ 𝑋 𝑦 = (π‘β€˜π‘₯)})
124, 6grpoinvcl 29508 . . . . . . 7 ((𝐺 ∈ GrpOp ∧ 𝑦 ∈ 𝑋) β†’ (π‘β€˜π‘¦) ∈ 𝑋)
134, 6grpo2inv 29515 . . . . . . . 8 ((𝐺 ∈ GrpOp ∧ 𝑦 ∈ 𝑋) β†’ (π‘β€˜(π‘β€˜π‘¦)) = 𝑦)
1413eqcomd 2739 . . . . . . 7 ((𝐺 ∈ GrpOp ∧ 𝑦 ∈ 𝑋) β†’ 𝑦 = (π‘β€˜(π‘β€˜π‘¦)))
15 fveq2 6843 . . . . . . . 8 (π‘₯ = (π‘β€˜π‘¦) β†’ (π‘β€˜π‘₯) = (π‘β€˜(π‘β€˜π‘¦)))
1615rspceeqv 3596 . . . . . . 7 (((π‘β€˜π‘¦) ∈ 𝑋 ∧ 𝑦 = (π‘β€˜(π‘β€˜π‘¦))) β†’ βˆƒπ‘₯ ∈ 𝑋 𝑦 = (π‘β€˜π‘₯))
1712, 14, 16syl2anc 585 . . . . . 6 ((𝐺 ∈ GrpOp ∧ 𝑦 ∈ 𝑋) β†’ βˆƒπ‘₯ ∈ 𝑋 𝑦 = (π‘β€˜π‘₯))
1817ex 414 . . . . 5 (𝐺 ∈ GrpOp β†’ (𝑦 ∈ 𝑋 β†’ βˆƒπ‘₯ ∈ 𝑋 𝑦 = (π‘β€˜π‘₯)))
19 simpr 486 . . . . . . 7 (((𝐺 ∈ GrpOp ∧ π‘₯ ∈ 𝑋) ∧ 𝑦 = (π‘β€˜π‘₯)) β†’ 𝑦 = (π‘β€˜π‘₯))
204, 6grpoinvcl 29508 . . . . . . . 8 ((𝐺 ∈ GrpOp ∧ π‘₯ ∈ 𝑋) β†’ (π‘β€˜π‘₯) ∈ 𝑋)
2120adantr 482 . . . . . . 7 (((𝐺 ∈ GrpOp ∧ π‘₯ ∈ 𝑋) ∧ 𝑦 = (π‘β€˜π‘₯)) β†’ (π‘β€˜π‘₯) ∈ 𝑋)
2219, 21eqeltrd 2834 . . . . . 6 (((𝐺 ∈ GrpOp ∧ π‘₯ ∈ 𝑋) ∧ 𝑦 = (π‘β€˜π‘₯)) β†’ 𝑦 ∈ 𝑋)
2322rexlimdva2 3151 . . . . 5 (𝐺 ∈ GrpOp β†’ (βˆƒπ‘₯ ∈ 𝑋 𝑦 = (π‘β€˜π‘₯) β†’ 𝑦 ∈ 𝑋))
2418, 23impbid 211 . . . 4 (𝐺 ∈ GrpOp β†’ (𝑦 ∈ 𝑋 ↔ βˆƒπ‘₯ ∈ 𝑋 𝑦 = (π‘β€˜π‘₯)))
2524abbi2dv 2868 . . 3 (𝐺 ∈ GrpOp β†’ 𝑋 = {𝑦 ∣ βˆƒπ‘₯ ∈ 𝑋 𝑦 = (π‘β€˜π‘₯)})
2611, 25eqtr4d 2776 . 2 (𝐺 ∈ GrpOp β†’ ran 𝑁 = 𝑋)
27 fveq2 6843 . . . 4 ((π‘β€˜π‘₯) = (π‘β€˜π‘¦) β†’ (π‘β€˜(π‘β€˜π‘₯)) = (π‘β€˜(π‘β€˜π‘¦)))
284, 6grpo2inv 29515 . . . . . 6 ((𝐺 ∈ GrpOp ∧ π‘₯ ∈ 𝑋) β†’ (π‘β€˜(π‘β€˜π‘₯)) = π‘₯)
2928, 13eqeqan12d 2747 . . . . 5 (((𝐺 ∈ GrpOp ∧ π‘₯ ∈ 𝑋) ∧ (𝐺 ∈ GrpOp ∧ 𝑦 ∈ 𝑋)) β†’ ((π‘β€˜(π‘β€˜π‘₯)) = (π‘β€˜(π‘β€˜π‘¦)) ↔ π‘₯ = 𝑦))
3029anandis 677 . . . 4 ((𝐺 ∈ GrpOp ∧ (π‘₯ ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) β†’ ((π‘β€˜(π‘β€˜π‘₯)) = (π‘β€˜(π‘β€˜π‘¦)) ↔ π‘₯ = 𝑦))
3127, 30imbitrid 243 . . 3 ((𝐺 ∈ GrpOp ∧ (π‘₯ ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) β†’ ((π‘β€˜π‘₯) = (π‘β€˜π‘¦) β†’ π‘₯ = 𝑦))
3231ralrimivva 3194 . 2 (𝐺 ∈ GrpOp β†’ βˆ€π‘₯ ∈ 𝑋 βˆ€π‘¦ ∈ 𝑋 ((π‘β€˜π‘₯) = (π‘β€˜π‘¦) β†’ π‘₯ = 𝑦))
33 dff1o6 7222 . 2 (𝑁:𝑋–1-1-onto→𝑋 ↔ (𝑁 Fn 𝑋 ∧ ran 𝑁 = 𝑋 ∧ βˆ€π‘₯ ∈ 𝑋 βˆ€π‘¦ ∈ 𝑋 ((π‘β€˜π‘₯) = (π‘β€˜π‘¦) β†’ π‘₯ = 𝑦)))
349, 26, 32, 33syl3anbrc 1344 1 (𝐺 ∈ GrpOp β†’ 𝑁:𝑋–1-1-onto→𝑋)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 397   = wceq 1542   ∈ wcel 2107  {cab 2710  βˆ€wral 3061  βˆƒwrex 3070   ↦ cmpt 5189  ran crn 5635   Fn wfn 6492  β€“1-1-ontoβ†’wf1o 6496  β€˜cfv 6497  β„©crio 7313  (class class class)co 7358  GrpOpcgr 29473  GIdcgi 29474  invcgn 29475
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5243  ax-sep 5257  ax-nul 5264  ax-pr 5385  ax-un 7673
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3353  df-rab 3407  df-v 3446  df-sbc 3741  df-csb 3857  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-iun 4957  df-br 5107  df-opab 5169  df-mpt 5190  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-f1 6502  df-fo 6503  df-f1o 6504  df-fv 6505  df-riota 7314  df-ov 7361  df-grpo 29477  df-gid 29478  df-ginv 29479
This theorem is referenced by:  nvinvfval  29624
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