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Theorem pmtrrn 17793
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 6439 . . . . . . 7 (𝐷𝑉 → (𝑦𝐷 ↦ if(𝑦𝑧, (𝑧 ∖ {𝑦}), 𝑦)) ∈ V)
21ralrimivw 2966 . . . . . 6 (𝐷𝑉 → ∀𝑧 ∈ {𝑥 ∈ 𝒫 𝐷𝑥 ≈ 2𝑜} (𝑦𝐷 ↦ if(𝑦𝑧, (𝑧 ∖ {𝑦}), 𝑦)) ∈ V)
323ad2ant1 1080 . . . . 5 ((𝐷𝑉𝑃𝐷𝑃 ≈ 2𝑜) → ∀𝑧 ∈ {𝑥 ∈ 𝒫 𝐷𝑥 ≈ 2𝑜} (𝑦𝐷 ↦ if(𝑦𝑧, (𝑧 ∖ {𝑦}), 𝑦)) ∈ V)
4 eqid 2626 . . . . . 6 (𝑧 ∈ {𝑥 ∈ 𝒫 𝐷𝑥 ≈ 2𝑜} ↦ (𝑦𝐷 ↦ if(𝑦𝑧, (𝑧 ∖ {𝑦}), 𝑦))) = (𝑧 ∈ {𝑥 ∈ 𝒫 𝐷𝑥 ≈ 2𝑜} ↦ (𝑦𝐷 ↦ if(𝑦𝑧, (𝑧 ∖ {𝑦}), 𝑦)))
54fnmpt 5979 . . . . 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 17786 . . . . . 6 (𝐷𝑉𝑇 = (𝑧 ∈ {𝑥 ∈ 𝒫 𝐷𝑥 ≈ 2𝑜} ↦ (𝑦𝐷 ↦ if(𝑦𝑧, (𝑧 ∖ {𝑦}), 𝑦))))
983ad2ant1 1080 . . . . 5 ((𝐷𝑉𝑃𝐷𝑃 ≈ 2𝑜) → 𝑇 = (𝑧 ∈ {𝑥 ∈ 𝒫 𝐷𝑥 ≈ 2𝑜} ↦ (𝑦𝐷 ↦ if(𝑦𝑧, (𝑧 ∖ {𝑦}), 𝑦))))
109fneq1d 5941 . . . 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 247 . . 3 ((𝐷𝑉𝑃𝐷𝑃 ≈ 2𝑜) → 𝑇 Fn {𝑥 ∈ 𝒫 𝐷𝑥 ≈ 2𝑜})
12 elpw2g 4792 . . . . . 6 (𝐷𝑉 → (𝑃 ∈ 𝒫 𝐷𝑃𝐷))
1312biimpar 502 . . . . 5 ((𝐷𝑉𝑃𝐷) → 𝑃 ∈ 𝒫 𝐷)
14133adant3 1079 . . . 4 ((𝐷𝑉𝑃𝐷𝑃 ≈ 2𝑜) → 𝑃 ∈ 𝒫 𝐷)
15 simp3 1061 . . . 4 ((𝐷𝑉𝑃𝐷𝑃 ≈ 2𝑜) → 𝑃 ≈ 2𝑜)
16 breq1 4621 . . . . 5 (𝑥 = 𝑃 → (𝑥 ≈ 2𝑜𝑃 ≈ 2𝑜))
1716elrab 3351 . . . 4 (𝑃 ∈ {𝑥 ∈ 𝒫 𝐷𝑥 ≈ 2𝑜} ↔ (𝑃 ∈ 𝒫 𝐷𝑃 ≈ 2𝑜))
1814, 15, 17sylanbrc 697 . . 3 ((𝐷𝑉𝑃𝐷𝑃 ≈ 2𝑜) → 𝑃 ∈ {𝑥 ∈ 𝒫 𝐷𝑥 ≈ 2𝑜})
19 fnfvelrn 6313 . . 3 ((𝑇 Fn {𝑥 ∈ 𝒫 𝐷𝑥 ≈ 2𝑜} ∧ 𝑃 ∈ {𝑥 ∈ 𝒫 𝐷𝑥 ≈ 2𝑜}) → (𝑇𝑃) ∈ ran 𝑇)
2011, 18, 19syl2anc 692 . 2 ((𝐷𝑉𝑃𝐷𝑃 ≈ 2𝑜) → (𝑇𝑃) ∈ ran 𝑇)
21 pmtrrn.r . 2 𝑅 = ran 𝑇
2220, 21syl6eleqr 2715 1 ((𝐷𝑉𝑃𝐷𝑃 ≈ 2𝑜) → (𝑇𝑃) ∈ 𝑅)
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1036   = wceq 1480  wcel 1992  wral 2912  {crab 2916  Vcvv 3191  cdif 3557  wss 3560  ifcif 4063  𝒫 cpw 4135  {csn 4153   cuni 4407   class class class wbr 4618  cmpt 4678  ran crn 5080   Fn wfn 5845  cfv 5850  2𝑜c2o 7500  cen 7897  pmTrspcpmtr 17777
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 1841  ax-6 1890  ax-7 1937  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606  ax-rep 4736  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872
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 1883  df-eu 2478  df-mo 2479  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ne 2797  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3193  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-id 4994  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-iota 5813  df-fun 5852  df-fn 5853  df-f 5854  df-f1 5855  df-fo 5856  df-f1o 5857  df-fv 5858  df-pmtr 17778
This theorem is referenced by:  pmtrfb  17801  symggen  17806  pmtr3ncom  17811  pmtrdifellem1  17812  mdetralt  20328  pmtrto1cl  29626
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