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Theorem pmtrrn 17998
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 6600 . . . . . . 7 (𝐷𝑉 → (𝑦𝐷 ↦ if(𝑦𝑧, (𝑧 ∖ {𝑦}), 𝑦)) ∈ V)
21ralrimivw 3069 . . . . . 6 (𝐷𝑉 → ∀𝑧 ∈ {𝑥 ∈ 𝒫 𝐷𝑥 ≈ 2𝑜} (𝑦𝐷 ↦ if(𝑦𝑧, (𝑧 ∖ {𝑦}), 𝑦)) ∈ V)
323ad2ant1 1125 . . . . 5 ((𝐷𝑉𝑃𝐷𝑃 ≈ 2𝑜) → ∀𝑧 ∈ {𝑥 ∈ 𝒫 𝐷𝑥 ≈ 2𝑜} (𝑦𝐷 ↦ if(𝑦𝑧, (𝑧 ∖ {𝑦}), 𝑦)) ∈ V)
4 eqid 2724 . . . . . 6 (𝑧 ∈ {𝑥 ∈ 𝒫 𝐷𝑥 ≈ 2𝑜} ↦ (𝑦𝐷 ↦ if(𝑦𝑧, (𝑧 ∖ {𝑦}), 𝑦))) = (𝑧 ∈ {𝑥 ∈ 𝒫 𝐷𝑥 ≈ 2𝑜} ↦ (𝑦𝐷 ↦ if(𝑦𝑧, (𝑧 ∖ {𝑦}), 𝑦)))
54fnmpt 6133 . . . . 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 17991 . . . . . 6 (𝐷𝑉𝑇 = (𝑧 ∈ {𝑥 ∈ 𝒫 𝐷𝑥 ≈ 2𝑜} ↦ (𝑦𝐷 ↦ if(𝑦𝑧, (𝑧 ∖ {𝑦}), 𝑦))))
983ad2ant1 1125 . . . . 5 ((𝐷𝑉𝑃𝐷𝑃 ≈ 2𝑜) → 𝑇 = (𝑧 ∈ {𝑥 ∈ 𝒫 𝐷𝑥 ≈ 2𝑜} ↦ (𝑦𝐷 ↦ if(𝑦𝑧, (𝑧 ∖ {𝑦}), 𝑦))))
109fneq1d 6094 . . . 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 4932 . . . . . 6 (𝐷𝑉 → (𝑃 ∈ 𝒫 𝐷𝑃𝐷))
1312biimpar 503 . . . . 5 ((𝐷𝑉𝑃𝐷) → 𝑃 ∈ 𝒫 𝐷)
14133adant3 1124 . . . 4 ((𝐷𝑉𝑃𝐷𝑃 ≈ 2𝑜) → 𝑃 ∈ 𝒫 𝐷)
15 simp3 1130 . . . 4 ((𝐷𝑉𝑃𝐷𝑃 ≈ 2𝑜) → 𝑃 ≈ 2𝑜)
16 breq1 4763 . . . . 5 (𝑥 = 𝑃 → (𝑥 ≈ 2𝑜𝑃 ≈ 2𝑜))
1716elrab 3469 . . . 4 (𝑃 ∈ {𝑥 ∈ 𝒫 𝐷𝑥 ≈ 2𝑜} ↔ (𝑃 ∈ 𝒫 𝐷𝑃 ≈ 2𝑜))
1814, 15, 17sylanbrc 701 . . 3 ((𝐷𝑉𝑃𝐷𝑃 ≈ 2𝑜) → 𝑃 ∈ {𝑥 ∈ 𝒫 𝐷𝑥 ≈ 2𝑜})
19 fnfvelrn 6471 . . 3 ((𝑇 Fn {𝑥 ∈ 𝒫 𝐷𝑥 ≈ 2𝑜} ∧ 𝑃 ∈ {𝑥 ∈ 𝒫 𝐷𝑥 ≈ 2𝑜}) → (𝑇𝑃) ∈ ran 𝑇)
2011, 18, 19syl2anc 696 . 2 ((𝐷𝑉𝑃𝐷𝑃 ≈ 2𝑜) → (𝑇𝑃) ∈ ran 𝑇)
21 pmtrrn.r . 2 𝑅 = ran 𝑇
2220, 21syl6eleqr 2814 1 ((𝐷𝑉𝑃𝐷𝑃 ≈ 2𝑜) → (𝑇𝑃) ∈ 𝑅)
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1072   = wceq 1596  wcel 2103  wral 3014  {crab 3018  Vcvv 3304  cdif 3677  wss 3680  ifcif 4194  𝒫 cpw 4266  {csn 4285   cuni 4544   class class class wbr 4760  cmpt 4837  ran crn 5219   Fn wfn 5996  cfv 6001  2𝑜c2o 7674  cen 8069  pmTrspcpmtr 17982
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-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-pow 4948  ax-pr 5011
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-pw 4268  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-pmtr 17983
This theorem is referenced by:  pmtrfb  18006  symggen  18011  pmtr3ncom  18016  pmtrdifellem1  18017  mdetralt  20537  pmtrto1cl  30079  pmtridf1o  30086
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