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Theorem pmtrrn 18142
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 6677 . . . . . . 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 1163 . . . . 5 ((𝐷𝑉𝑃𝐷𝑃 ≈ 2𝑜) → ∀𝑧 ∈ {𝑥 ∈ 𝒫 𝐷𝑥 ≈ 2𝑜} (𝑦𝐷 ↦ if(𝑦𝑧, (𝑧 ∖ {𝑦}), 𝑦)) ∈ V)
4 eqid 2765 . . . . . 6 (𝑧 ∈ {𝑥 ∈ 𝒫 𝐷𝑥 ≈ 2𝑜} ↦ (𝑦𝐷 ↦ if(𝑦𝑧, (𝑧 ∖ {𝑦}), 𝑦))) = (𝑧 ∈ {𝑥 ∈ 𝒫 𝐷𝑥 ≈ 2𝑜} ↦ (𝑦𝐷 ↦ if(𝑦𝑧, (𝑧 ∖ {𝑦}), 𝑦)))
54fnmpt 6198 . . . . 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 18135 . . . . . 6 (𝐷𝑉𝑇 = (𝑧 ∈ {𝑥 ∈ 𝒫 𝐷𝑥 ≈ 2𝑜} ↦ (𝑦𝐷 ↦ if(𝑦𝑧, (𝑧 ∖ {𝑦}), 𝑦))))
983ad2ant1 1163 . . . . 5 ((𝐷𝑉𝑃𝐷𝑃 ≈ 2𝑜) → 𝑇 = (𝑧 ∈ {𝑥 ∈ 𝒫 𝐷𝑥 ≈ 2𝑜} ↦ (𝑦𝐷 ↦ if(𝑦𝑧, (𝑧 ∖ {𝑦}), 𝑦))))
109fneq1d 6159 . . . 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 248 . . 3 ((𝐷𝑉𝑃𝐷𝑃 ≈ 2𝑜) → 𝑇 Fn {𝑥 ∈ 𝒫 𝐷𝑥 ≈ 2𝑜})
12 elpw2g 4985 . . . . . 6 (𝐷𝑉 → (𝑃 ∈ 𝒫 𝐷𝑃𝐷))
1312biimpar 469 . . . . 5 ((𝐷𝑉𝑃𝐷) → 𝑃 ∈ 𝒫 𝐷)
14133adant3 1162 . . . 4 ((𝐷𝑉𝑃𝐷𝑃 ≈ 2𝑜) → 𝑃 ∈ 𝒫 𝐷)
15 simp3 1168 . . . 4 ((𝐷𝑉𝑃𝐷𝑃 ≈ 2𝑜) → 𝑃 ≈ 2𝑜)
16 breq1 4812 . . . . 5 (𝑥 = 𝑃 → (𝑥 ≈ 2𝑜𝑃 ≈ 2𝑜))
1716elrab 3519 . . . 4 (𝑃 ∈ {𝑥 ∈ 𝒫 𝐷𝑥 ≈ 2𝑜} ↔ (𝑃 ∈ 𝒫 𝐷𝑃 ≈ 2𝑜))
1814, 15, 17sylanbrc 578 . . 3 ((𝐷𝑉𝑃𝐷𝑃 ≈ 2𝑜) → 𝑃 ∈ {𝑥 ∈ 𝒫 𝐷𝑥 ≈ 2𝑜})
19 fnfvelrn 6546 . . 3 ((𝑇 Fn {𝑥 ∈ 𝒫 𝐷𝑥 ≈ 2𝑜} ∧ 𝑃 ∈ {𝑥 ∈ 𝒫 𝐷𝑥 ≈ 2𝑜}) → (𝑇𝑃) ∈ ran 𝑇)
2011, 18, 19syl2anc 579 . 2 ((𝐷𝑉𝑃𝐷𝑃 ≈ 2𝑜) → (𝑇𝑃) ∈ ran 𝑇)
21 pmtrrn.r . 2 𝑅 = ran 𝑇
2220, 21syl6eleqr 2855 1 ((𝐷𝑉𝑃𝐷𝑃 ≈ 2𝑜) → (𝑇𝑃) ∈ 𝑅)
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1107   = wceq 1652  wcel 2155  wral 3055  {crab 3059  Vcvv 3350  cdif 3729  wss 3732  ifcif 4243  𝒫 cpw 4315  {csn 4334   cuni 4594   class class class wbr 4809  cmpt 4888  ran crn 5278   Fn wfn 6063  cfv 6068  2𝑜c2o 7758  cen 8157  pmTrspcpmtr 18126
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-rep 4930  ax-sep 4941  ax-nul 4949  ax-pow 5001  ax-pr 5062
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-ral 3060  df-rex 3061  df-reu 3062  df-rab 3064  df-v 3352  df-sbc 3597  df-csb 3692  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-nul 4080  df-if 4244  df-pw 4317  df-sn 4335  df-pr 4337  df-op 4341  df-uni 4595  df-iun 4678  df-br 4810  df-opab 4872  df-mpt 4889  df-id 5185  df-xp 5283  df-rel 5284  df-cnv 5285  df-co 5286  df-dm 5287  df-rn 5288  df-res 5289  df-ima 5290  df-iota 6031  df-fun 6070  df-fn 6071  df-f 6072  df-f1 6073  df-fo 6074  df-f1o 6075  df-fv 6076  df-pmtr 18127
This theorem is referenced by:  pmtrfb  18150  symggen  18155  pmtr3ncom  18160  pmtrdifellem1  18161  mdetralt  20691  pmtrto1cl  30231  pmtridf1o  30238
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