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Theorem pmtrfval 19318
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 3493 . . 3 (𝐷𝑉𝐷 ∈ V)
3 pweq 4617 . . . . . 6 (𝑑 = 𝐷 → 𝒫 𝑑 = 𝒫 𝐷)
43rabeqdv 3448 . . . . 5 (𝑑 = 𝐷 → {𝑦 ∈ 𝒫 𝑑𝑦 ≈ 2o} = {𝑦 ∈ 𝒫 𝐷𝑦 ≈ 2o})
5 mpteq1 5242 . . . . 5 (𝑑 = 𝐷 → (𝑧𝑑 ↦ if(𝑧𝑝, (𝑝 ∖ {𝑧}), 𝑧)) = (𝑧𝐷 ↦ if(𝑧𝑝, (𝑝 ∖ {𝑧}), 𝑧)))
64, 5mpteq12dv 5240 . . . 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 19310 . . . 4 pmTrsp = (𝑑 ∈ V ↦ (𝑝 ∈ {𝑦 ∈ 𝒫 𝑑𝑦 ≈ 2o} ↦ (𝑧𝑑 ↦ if(𝑧𝑝, (𝑝 ∖ {𝑧}), 𝑧))))
8 vpwex 5376 . . . . 5 𝒫 𝑑 ∈ V
98mptrabex 7227 . . . 4 (𝑝 ∈ {𝑦 ∈ 𝒫 𝑑𝑦 ≈ 2o} ↦ (𝑧𝑑 ↦ if(𝑧𝑝, (𝑝 ∖ {𝑧}), 𝑧))) ∈ V
106, 7, 9fvmpt3i 7004 . . 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 2785 1 (𝐷𝑉𝑇 = (𝑝 ∈ {𝑦 ∈ 𝒫 𝐷𝑦 ≈ 2o} ↦ (𝑧𝐷 ↦ if(𝑧𝑝, (𝑝 ∖ {𝑧}), 𝑧))))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1542  wcel 2107  {crab 3433  Vcvv 3475  cdif 3946  ifcif 4529  𝒫 cpw 4603  {csn 4629   cuni 4909   class class class wbr 5149  cmpt 5232  cfv 6544  2oc2o 8460  cen 8936  pmTrspcpmtr 19309
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-pmtr 19310
This theorem is referenced by:  pmtrval  19319  pmtrrn  19325  pmtrfrn  19326  pmtrprfval  19355  pmtrsn  19387  trsp2cyc  32282
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