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Theorem pmtrrn 18652
 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 6975 . . . . . . 7 (𝐷𝑉 → (𝑦𝐷 ↦ if(𝑦𝑧, (𝑧 ∖ {𝑦}), 𝑦)) ∈ V)
21ralrimivw 3114 . . . . . 6 (𝐷𝑉 → ∀𝑧 ∈ {𝑥 ∈ 𝒫 𝐷𝑥 ≈ 2o} (𝑦𝐷 ↦ if(𝑦𝑧, (𝑧 ∖ {𝑦}), 𝑦)) ∈ V)
323ad2ant1 1130 . . . . 5 ((𝐷𝑉𝑃𝐷𝑃 ≈ 2o) → ∀𝑧 ∈ {𝑥 ∈ 𝒫 𝐷𝑥 ≈ 2o} (𝑦𝐷 ↦ if(𝑦𝑧, (𝑧 ∖ {𝑦}), 𝑦)) ∈ V)
4 eqid 2758 . . . . . 6 (𝑧 ∈ {𝑥 ∈ 𝒫 𝐷𝑥 ≈ 2o} ↦ (𝑦𝐷 ↦ if(𝑦𝑧, (𝑧 ∖ {𝑦}), 𝑦))) = (𝑧 ∈ {𝑥 ∈ 𝒫 𝐷𝑥 ≈ 2o} ↦ (𝑦𝐷 ↦ if(𝑦𝑧, (𝑧 ∖ {𝑦}), 𝑦)))
54fnmpt 6471 . . . . 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 18645 . . . . . 6 (𝐷𝑉𝑇 = (𝑧 ∈ {𝑥 ∈ 𝒫 𝐷𝑥 ≈ 2o} ↦ (𝑦𝐷 ↦ if(𝑦𝑧, (𝑧 ∖ {𝑦}), 𝑦))))
983ad2ant1 1130 . . . . 5 ((𝐷𝑉𝑃𝐷𝑃 ≈ 2o) → 𝑇 = (𝑧 ∈ {𝑥 ∈ 𝒫 𝐷𝑥 ≈ 2o} ↦ (𝑦𝐷 ↦ if(𝑦𝑧, (𝑧 ∖ {𝑦}), 𝑦))))
109fneq1d 6427 . . . 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 260 . . 3 ((𝐷𝑉𝑃𝐷𝑃 ≈ 2o) → 𝑇 Fn {𝑥 ∈ 𝒫 𝐷𝑥 ≈ 2o})
12 breq1 5035 . . . 4 (𝑥 = 𝑃 → (𝑥 ≈ 2o𝑃 ≈ 2o))
13 elpw2g 5214 . . . . . 6 (𝐷𝑉 → (𝑃 ∈ 𝒫 𝐷𝑃𝐷))
1413biimpar 481 . . . . 5 ((𝐷𝑉𝑃𝐷) → 𝑃 ∈ 𝒫 𝐷)
15143adant3 1129 . . . 4 ((𝐷𝑉𝑃𝐷𝑃 ≈ 2o) → 𝑃 ∈ 𝒫 𝐷)
16 simp3 1135 . . . 4 ((𝐷𝑉𝑃𝐷𝑃 ≈ 2o) → 𝑃 ≈ 2o)
1712, 15, 16elrabd 3604 . . 3 ((𝐷𝑉𝑃𝐷𝑃 ≈ 2o) → 𝑃 ∈ {𝑥 ∈ 𝒫 𝐷𝑥 ≈ 2o})
18 fnfvelrn 6839 . . 3 ((𝑇 Fn {𝑥 ∈ 𝒫 𝐷𝑥 ≈ 2o} ∧ 𝑃 ∈ {𝑥 ∈ 𝒫 𝐷𝑥 ≈ 2o}) → (𝑇𝑃) ∈ ran 𝑇)
1911, 17, 18syl2anc 587 . 2 ((𝐷𝑉𝑃𝐷𝑃 ≈ 2o) → (𝑇𝑃) ∈ ran 𝑇)
20 pmtrrn.r . 2 𝑅 = ran 𝑇
2119, 20eleqtrrdi 2863 1 ((𝐷𝑉𝑃𝐷𝑃 ≈ 2o) → (𝑇𝑃) ∈ 𝑅)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ w3a 1084   = wceq 1538   ∈ wcel 2111  ∀wral 3070  {crab 3074  Vcvv 3409   ∖ cdif 3855   ⊆ wss 3858  ifcif 4420  𝒫 cpw 4494  {csn 4522  ∪ cuni 4798   class class class wbr 5032   ↦ cmpt 5112  ran crn 5525   Fn wfn 6330  ‘cfv 6335  2oc2o 8106   ≈ cen 8524  pmTrspcpmtr 18636 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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-rep 5156  ax-sep 5169  ax-nul 5176  ax-pow 5234  ax-pr 5298 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ne 2952  df-ral 3075  df-rex 3076  df-reu 3077  df-rab 3079  df-v 3411  df-sbc 3697  df-csb 3806  df-dif 3861  df-un 3863  df-in 3865  df-ss 3875  df-nul 4226  df-if 4421  df-pw 4496  df-sn 4523  df-pr 4525  df-op 4529  df-uni 4799  df-iun 4885  df-br 5033  df-opab 5095  df-mpt 5113  df-id 5430  df-xp 5530  df-rel 5531  df-cnv 5532  df-co 5533  df-dm 5534  df-rn 5535  df-res 5536  df-ima 5537  df-iota 6294  df-fun 6337  df-fn 6338  df-f 6339  df-f1 6340  df-fo 6341  df-f1o 6342  df-fv 6343  df-pmtr 18637 This theorem is referenced by:  pmtrfb  18660  symggen  18665  pmtr3ncom  18670  pmtrdifellem1  18671  mdetralt  21308  pmtrcnel  30884  pmtrcnel2  30885  pmtridf1o  30887  pmtrto1cl  30892  cyc3evpm  30943  cyc3genpmlem  30944  cyc3conja  30950
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