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Theorem pmtrrn 19230
Description: Transposing two points gives a transposition function. (Contributed by Stefan O'Rear, 22-Aug-2015.)
Hypotheses
Ref Expression
pmtrrn.t 𝑇 = (pmTrsp‘𝐷)
pmtrrn.r 𝑅 = ran 𝑇
Assertion
Ref Expression
pmtrrn ((𝐷𝑉𝑃𝐷𝑃 ≈ 2o) → (𝑇𝑃) ∈ 𝑅)

Proof of Theorem pmtrrn
Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 mptexg 7167 . . . . . . 7 (𝐷𝑉 → (𝑦𝐷 ↦ if(𝑦𝑧, (𝑧 ∖ {𝑦}), 𝑦)) ∈ V)
21ralrimivw 3145 . . . . . 6 (𝐷𝑉 → ∀𝑧 ∈ {𝑥 ∈ 𝒫 𝐷𝑥 ≈ 2o} (𝑦𝐷 ↦ if(𝑦𝑧, (𝑧 ∖ {𝑦}), 𝑦)) ∈ V)
323ad2ant1 1133 . . . . 5 ((𝐷𝑉𝑃𝐷𝑃 ≈ 2o) → ∀𝑧 ∈ {𝑥 ∈ 𝒫 𝐷𝑥 ≈ 2o} (𝑦𝐷 ↦ if(𝑦𝑧, (𝑧 ∖ {𝑦}), 𝑦)) ∈ V)
4 eqid 2736 . . . . . 6 (𝑧 ∈ {𝑥 ∈ 𝒫 𝐷𝑥 ≈ 2o} ↦ (𝑦𝐷 ↦ if(𝑦𝑧, (𝑧 ∖ {𝑦}), 𝑦))) = (𝑧 ∈ {𝑥 ∈ 𝒫 𝐷𝑥 ≈ 2o} ↦ (𝑦𝐷 ↦ if(𝑦𝑧, (𝑧 ∖ {𝑦}), 𝑦)))
54fnmpt 6638 . . . . 5 (∀𝑧 ∈ {𝑥 ∈ 𝒫 𝐷𝑥 ≈ 2o} (𝑦𝐷 ↦ if(𝑦𝑧, (𝑧 ∖ {𝑦}), 𝑦)) ∈ V → (𝑧 ∈ {𝑥 ∈ 𝒫 𝐷𝑥 ≈ 2o} ↦ (𝑦𝐷 ↦ if(𝑦𝑧, (𝑧 ∖ {𝑦}), 𝑦))) Fn {𝑥 ∈ 𝒫 𝐷𝑥 ≈ 2o})
63, 5syl 17 . . . 4 ((𝐷𝑉𝑃𝐷𝑃 ≈ 2o) → (𝑧 ∈ {𝑥 ∈ 𝒫 𝐷𝑥 ≈ 2o} ↦ (𝑦𝐷 ↦ if(𝑦𝑧, (𝑧 ∖ {𝑦}), 𝑦))) Fn {𝑥 ∈ 𝒫 𝐷𝑥 ≈ 2o})
7 pmtrrn.t . . . . . . 7 𝑇 = (pmTrsp‘𝐷)
87pmtrfval 19223 . . . . . 6 (𝐷𝑉𝑇 = (𝑧 ∈ {𝑥 ∈ 𝒫 𝐷𝑥 ≈ 2o} ↦ (𝑦𝐷 ↦ if(𝑦𝑧, (𝑧 ∖ {𝑦}), 𝑦))))
983ad2ant1 1133 . . . . 5 ((𝐷𝑉𝑃𝐷𝑃 ≈ 2o) → 𝑇 = (𝑧 ∈ {𝑥 ∈ 𝒫 𝐷𝑥 ≈ 2o} ↦ (𝑦𝐷 ↦ if(𝑦𝑧, (𝑧 ∖ {𝑦}), 𝑦))))
109fneq1d 6592 . . . 4 ((𝐷𝑉𝑃𝐷𝑃 ≈ 2o) → (𝑇 Fn {𝑥 ∈ 𝒫 𝐷𝑥 ≈ 2o} ↔ (𝑧 ∈ {𝑥 ∈ 𝒫 𝐷𝑥 ≈ 2o} ↦ (𝑦𝐷 ↦ if(𝑦𝑧, (𝑧 ∖ {𝑦}), 𝑦))) Fn {𝑥 ∈ 𝒫 𝐷𝑥 ≈ 2o}))
116, 10mpbird 256 . . 3 ((𝐷𝑉𝑃𝐷𝑃 ≈ 2o) → 𝑇 Fn {𝑥 ∈ 𝒫 𝐷𝑥 ≈ 2o})
12 breq1 5106 . . . 4 (𝑥 = 𝑃 → (𝑥 ≈ 2o𝑃 ≈ 2o))
13 elpw2g 5299 . . . . . 6 (𝐷𝑉 → (𝑃 ∈ 𝒫 𝐷𝑃𝐷))
1413biimpar 478 . . . . 5 ((𝐷𝑉𝑃𝐷) → 𝑃 ∈ 𝒫 𝐷)
15143adant3 1132 . . . 4 ((𝐷𝑉𝑃𝐷𝑃 ≈ 2o) → 𝑃 ∈ 𝒫 𝐷)
16 simp3 1138 . . . 4 ((𝐷𝑉𝑃𝐷𝑃 ≈ 2o) → 𝑃 ≈ 2o)
1712, 15, 16elrabd 3645 . . 3 ((𝐷𝑉𝑃𝐷𝑃 ≈ 2o) → 𝑃 ∈ {𝑥 ∈ 𝒫 𝐷𝑥 ≈ 2o})
18 fnfvelrn 7028 . . 3 ((𝑇 Fn {𝑥 ∈ 𝒫 𝐷𝑥 ≈ 2o} ∧ 𝑃 ∈ {𝑥 ∈ 𝒫 𝐷𝑥 ≈ 2o}) → (𝑇𝑃) ∈ ran 𝑇)
1911, 17, 18syl2anc 584 . 2 ((𝐷𝑉𝑃𝐷𝑃 ≈ 2o) → (𝑇𝑃) ∈ ran 𝑇)
20 pmtrrn.r . 2 𝑅 = ran 𝑇
2119, 20eleqtrrdi 2849 1 ((𝐷𝑉𝑃𝐷𝑃 ≈ 2o) → (𝑇𝑃) ∈ 𝑅)
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1087   = wceq 1541  wcel 2106  wral 3062  {crab 3405  Vcvv 3443  cdif 3905  wss 3908  ifcif 4484  𝒫 cpw 4558  {csn 4584   cuni 4863   class class class wbr 5103  cmpt 5186  ran crn 5632   Fn wfn 6488  cfv 6493  2oc2o 8402  cen 8876  pmTrspcpmtr 19214
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5240  ax-sep 5254  ax-nul 5261  ax-pow 5318  ax-pr 5382
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2887  df-ne 2942  df-ral 3063  df-rex 3072  df-reu 3352  df-rab 3406  df-v 3445  df-sbc 3738  df-csb 3854  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4281  df-if 4485  df-pw 4560  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4864  df-iun 4954  df-br 5104  df-opab 5166  df-mpt 5187  df-id 5529  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-iota 6445  df-fun 6495  df-fn 6496  df-f 6497  df-f1 6498  df-fo 6499  df-f1o 6500  df-fv 6501  df-pmtr 19215
This theorem is referenced by:  pmtrfb  19238  symggen  19243  pmtr3ncom  19248  pmtrdifellem1  19249  mdetralt  21941  pmtrcnel  31823  pmtrcnel2  31824  pmtridf1o  31826  pmtrto1cl  31831  cyc3evpm  31882  cyc3genpmlem  31883  cyc3conja  31889
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