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Theorem pmtrfval 18220
Description: The function generating transpositions on a set. (Contributed by Stefan O'Rear, 16-Aug-2015.)
Hypothesis
Ref Expression
pmtrfval.t 𝑇 = (pmTrsp‘𝐷)
Assertion
Ref Expression
pmtrfval (𝐷𝑉𝑇 = (𝑝 ∈ {𝑦 ∈ 𝒫 𝐷𝑦 ≈ 2o} ↦ (𝑧𝐷 ↦ if(𝑧𝑝, (𝑝 ∖ {𝑧}), 𝑧))))
Distinct variable groups:   𝑦,𝑝,𝑧,𝐷   𝑇,𝑝,𝑦,𝑧   𝑧,𝑉
Allowed substitution hints:   𝑉(𝑦,𝑝)

Proof of Theorem pmtrfval
Dummy variable 𝑑 is distinct from all other variables.
StepHypRef Expression
1 pmtrfval.t . 2 𝑇 = (pmTrsp‘𝐷)
2 elex 3429 . . 3 (𝐷𝑉𝐷 ∈ V)
3 pweq 4381 . . . . . 6 (𝑑 = 𝐷 → 𝒫 𝑑 = 𝒫 𝐷)
43rabeqdv 3407 . . . . 5 (𝑑 = 𝐷 → {𝑦 ∈ 𝒫 𝑑𝑦 ≈ 2o} = {𝑦 ∈ 𝒫 𝐷𝑦 ≈ 2o})
5 mpteq1 4960 . . . . 5 (𝑑 = 𝐷 → (𝑧𝑑 ↦ if(𝑧𝑝, (𝑝 ∖ {𝑧}), 𝑧)) = (𝑧𝐷 ↦ if(𝑧𝑝, (𝑝 ∖ {𝑧}), 𝑧)))
64, 5mpteq12dv 4956 . . . 4 (𝑑 = 𝐷 → (𝑝 ∈ {𝑦 ∈ 𝒫 𝑑𝑦 ≈ 2o} ↦ (𝑧𝑑 ↦ if(𝑧𝑝, (𝑝 ∖ {𝑧}), 𝑧))) = (𝑝 ∈ {𝑦 ∈ 𝒫 𝐷𝑦 ≈ 2o} ↦ (𝑧𝐷 ↦ if(𝑧𝑝, (𝑝 ∖ {𝑧}), 𝑧))))
7 df-pmtr 18212 . . . 4 pmTrsp = (𝑑 ∈ V ↦ (𝑝 ∈ {𝑦 ∈ 𝒫 𝑑𝑦 ≈ 2o} ↦ (𝑧𝑑 ↦ if(𝑧𝑝, (𝑝 ∖ {𝑧}), 𝑧))))
8 vpwex 5077 . . . . 5 𝒫 𝑑 ∈ V
98mptrabex 6744 . . . 4 (𝑝 ∈ {𝑦 ∈ 𝒫 𝑑𝑦 ≈ 2o} ↦ (𝑧𝑑 ↦ if(𝑧𝑝, (𝑝 ∖ {𝑧}), 𝑧))) ∈ V
106, 7, 9fvmpt3i 6534 . . 3 (𝐷 ∈ V → (pmTrsp‘𝐷) = (𝑝 ∈ {𝑦 ∈ 𝒫 𝐷𝑦 ≈ 2o} ↦ (𝑧𝐷 ↦ if(𝑧𝑝, (𝑝 ∖ {𝑧}), 𝑧))))
112, 10syl 17 . 2 (𝐷𝑉 → (pmTrsp‘𝐷) = (𝑝 ∈ {𝑦 ∈ 𝒫 𝐷𝑦 ≈ 2o} ↦ (𝑧𝐷 ↦ if(𝑧𝑝, (𝑝 ∖ {𝑧}), 𝑧))))
121, 11syl5eq 2873 1 (𝐷𝑉𝑇 = (𝑝 ∈ {𝑦 ∈ 𝒫 𝐷𝑦 ≈ 2o} ↦ (𝑧𝐷 ↦ if(𝑧𝑝, (𝑝 ∖ {𝑧}), 𝑧))))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1656  wcel 2164  {crab 3121  Vcvv 3414  cdif 3795  ifcif 4306  𝒫 cpw 4378  {csn 4397   cuni 4658   class class class wbr 4873  cmpt 4952  cfv 6123  2oc2o 7820  cen 8219  pmTrspcpmtr 18211
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803  ax-rep 4994  ax-sep 5005  ax-nul 5013  ax-pow 5065  ax-pr 5127
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-3an 1113  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ne 3000  df-ral 3122  df-rex 3123  df-reu 3124  df-rab 3126  df-v 3416  df-sbc 3663  df-csb 3758  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4145  df-if 4307  df-pw 4380  df-sn 4398  df-pr 4400  df-op 4404  df-uni 4659  df-iun 4742  df-br 4874  df-opab 4936  df-mpt 4953  df-id 5250  df-xp 5348  df-rel 5349  df-cnv 5350  df-co 5351  df-dm 5352  df-rn 5353  df-res 5354  df-ima 5355  df-iota 6086  df-fun 6125  df-fn 6126  df-f 6127  df-f1 6128  df-fo 6129  df-f1o 6130  df-fv 6131  df-pmtr 18212
This theorem is referenced by:  pmtrval  18221  pmtrrn  18227  pmtrfrn  18228  pmtrprfval  18257  pmtrsn  18290
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