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Theorem pmtrfval 19359
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 3491 . . 3 (𝐷𝑉𝐷 ∈ V)
3 pweq 4615 . . . . . 6 (𝑑 = 𝐷 → 𝒫 𝑑 = 𝒫 𝐷)
43rabeqdv 3445 . . . . 5 (𝑑 = 𝐷 → {𝑦 ∈ 𝒫 𝑑𝑦 ≈ 2o} = {𝑦 ∈ 𝒫 𝐷𝑦 ≈ 2o})
5 mpteq1 5240 . . . . 5 (𝑑 = 𝐷 → (𝑧𝑑 ↦ if(𝑧𝑝, (𝑝 ∖ {𝑧}), 𝑧)) = (𝑧𝐷 ↦ if(𝑧𝑝, (𝑝 ∖ {𝑧}), 𝑧)))
64, 5mpteq12dv 5238 . . . 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 19351 . . . 4 pmTrsp = (𝑑 ∈ V ↦ (𝑝 ∈ {𝑦 ∈ 𝒫 𝑑𝑦 ≈ 2o} ↦ (𝑧𝑑 ↦ if(𝑧𝑝, (𝑝 ∖ {𝑧}), 𝑧))))
8 vpwex 5374 . . . . 5 𝒫 𝑑 ∈ V
98mptrabex 7228 . . . 4 (𝑝 ∈ {𝑦 ∈ 𝒫 𝑑𝑦 ≈ 2o} ↦ (𝑧𝑑 ↦ if(𝑧𝑝, (𝑝 ∖ {𝑧}), 𝑧))) ∈ V
106, 7, 9fvmpt3i 7002 . . 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, 11eqtrid 2782 1 (𝐷𝑉𝑇 = (𝑝 ∈ {𝑦 ∈ 𝒫 𝐷𝑦 ≈ 2o} ↦ (𝑧𝐷 ↦ if(𝑧𝑝, (𝑝 ∖ {𝑧}), 𝑧))))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1539  wcel 2104  {crab 3430  Vcvv 3472  cdif 3944  ifcif 4527  𝒫 cpw 4601  {csn 4627   cuni 4907   class class class wbr 5147  cmpt 5230  cfv 6542  2oc2o 8462  cen 8938  pmTrspcpmtr 19350
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3375  df-rab 3431  df-v 3474  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-pmtr 19351
This theorem is referenced by:  pmtrval  19360  pmtrrn  19366  pmtrfrn  19367  pmtrprfval  19396  pmtrsn  19428  trsp2cyc  32552
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