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Theorem pmtrval 18581
Description: A generated transposition, expressed in a symmetric form. (Contributed by Stefan O'Rear, 16-Aug-2015.)
Hypothesis
Ref Expression
pmtrfval.t 𝑇 = (pmTrsp‘𝐷)
Assertion
Ref Expression
pmtrval ((𝐷𝑉𝑃𝐷𝑃 ≈ 2o) → (𝑇𝑃) = (𝑧𝐷 ↦ if(𝑧𝑃, (𝑃 ∖ {𝑧}), 𝑧)))
Distinct variable groups:   𝑧,𝐷   𝑧,𝑇   𝑧,𝑃   𝑧,𝑉

Proof of Theorem pmtrval
Dummy variables 𝑝 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 pmtrfval.t . . . . 5 𝑇 = (pmTrsp‘𝐷)
21pmtrfval 18580 . . . 4 (𝐷𝑉𝑇 = (𝑝 ∈ {𝑦 ∈ 𝒫 𝐷𝑦 ≈ 2o} ↦ (𝑧𝐷 ↦ if(𝑧𝑝, (𝑝 ∖ {𝑧}), 𝑧))))
32fveq1d 6665 . . 3 (𝐷𝑉 → (𝑇𝑃) = ((𝑝 ∈ {𝑦 ∈ 𝒫 𝐷𝑦 ≈ 2o} ↦ (𝑧𝐷 ↦ if(𝑧𝑝, (𝑝 ∖ {𝑧}), 𝑧)))‘𝑃))
433ad2ant1 1130 . 2 ((𝐷𝑉𝑃𝐷𝑃 ≈ 2o) → (𝑇𝑃) = ((𝑝 ∈ {𝑦 ∈ 𝒫 𝐷𝑦 ≈ 2o} ↦ (𝑧𝐷 ↦ if(𝑧𝑝, (𝑝 ∖ {𝑧}), 𝑧)))‘𝑃))
5 eqid 2824 . . 3 (𝑝 ∈ {𝑦 ∈ 𝒫 𝐷𝑦 ≈ 2o} ↦ (𝑧𝐷 ↦ if(𝑧𝑝, (𝑝 ∖ {𝑧}), 𝑧))) = (𝑝 ∈ {𝑦 ∈ 𝒫 𝐷𝑦 ≈ 2o} ↦ (𝑧𝐷 ↦ if(𝑧𝑝, (𝑝 ∖ {𝑧}), 𝑧)))
6 eleq2 2904 . . . . 5 (𝑝 = 𝑃 → (𝑧𝑝𝑧𝑃))
7 difeq1 4078 . . . . . 6 (𝑝 = 𝑃 → (𝑝 ∖ {𝑧}) = (𝑃 ∖ {𝑧}))
87unieqd 4838 . . . . 5 (𝑝 = 𝑃 (𝑝 ∖ {𝑧}) = (𝑃 ∖ {𝑧}))
96, 8ifbieq1d 4473 . . . 4 (𝑝 = 𝑃 → if(𝑧𝑝, (𝑝 ∖ {𝑧}), 𝑧) = if(𝑧𝑃, (𝑃 ∖ {𝑧}), 𝑧))
109mpteq2dv 5149 . . 3 (𝑝 = 𝑃 → (𝑧𝐷 ↦ if(𝑧𝑝, (𝑝 ∖ {𝑧}), 𝑧)) = (𝑧𝐷 ↦ if(𝑧𝑃, (𝑃 ∖ {𝑧}), 𝑧)))
11 breq1 5056 . . . 4 (𝑦 = 𝑃 → (𝑦 ≈ 2o𝑃 ≈ 2o))
12 elpw2g 5234 . . . . . 6 (𝐷𝑉 → (𝑃 ∈ 𝒫 𝐷𝑃𝐷))
1312biimpar 481 . . . . 5 ((𝐷𝑉𝑃𝐷) → 𝑃 ∈ 𝒫 𝐷)
14133adant3 1129 . . . 4 ((𝐷𝑉𝑃𝐷𝑃 ≈ 2o) → 𝑃 ∈ 𝒫 𝐷)
15 simp3 1135 . . . 4 ((𝐷𝑉𝑃𝐷𝑃 ≈ 2o) → 𝑃 ≈ 2o)
1611, 14, 15elrabd 3668 . . 3 ((𝐷𝑉𝑃𝐷𝑃 ≈ 2o) → 𝑃 ∈ {𝑦 ∈ 𝒫 𝐷𝑦 ≈ 2o})
17 mptexg 6977 . . . 4 (𝐷𝑉 → (𝑧𝐷 ↦ if(𝑧𝑃, (𝑃 ∖ {𝑧}), 𝑧)) ∈ V)
18173ad2ant1 1130 . . 3 ((𝐷𝑉𝑃𝐷𝑃 ≈ 2o) → (𝑧𝐷 ↦ if(𝑧𝑃, (𝑃 ∖ {𝑧}), 𝑧)) ∈ V)
195, 10, 16, 18fvmptd3 6784 . 2 ((𝐷𝑉𝑃𝐷𝑃 ≈ 2o) → ((𝑝 ∈ {𝑦 ∈ 𝒫 𝐷𝑦 ≈ 2o} ↦ (𝑧𝐷 ↦ if(𝑧𝑝, (𝑝 ∖ {𝑧}), 𝑧)))‘𝑃) = (𝑧𝐷 ↦ if(𝑧𝑃, (𝑃 ∖ {𝑧}), 𝑧)))
204, 19eqtrd 2859 1 ((𝐷𝑉𝑃𝐷𝑃 ≈ 2o) → (𝑇𝑃) = (𝑧𝐷 ↦ if(𝑧𝑃, (𝑃 ∖ {𝑧}), 𝑧)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1084   = wceq 1538  wcel 2115  {crab 3137  Vcvv 3480  cdif 3916  wss 3919  ifcif 4450  𝒫 cpw 4522  {csn 4550   cuni 4824   class class class wbr 5053  cmpt 5133  cfv 6345  2oc2o 8094  cen 8504  pmTrspcpmtr 18571
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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-rep 5177  ax-sep 5190  ax-nul 5197  ax-pow 5254  ax-pr 5318
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-ral 3138  df-rex 3139  df-reu 3140  df-rab 3142  df-v 3482  df-sbc 3759  df-csb 3867  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-if 4451  df-pw 4524  df-sn 4551  df-pr 4553  df-op 4557  df-uni 4825  df-iun 4907  df-br 5054  df-opab 5116  df-mpt 5134  df-id 5448  df-xp 5549  df-rel 5550  df-cnv 5551  df-co 5552  df-dm 5553  df-rn 5554  df-res 5555  df-ima 5556  df-iota 6304  df-fun 6347  df-fn 6348  df-f 6349  df-f1 6350  df-fo 6351  df-f1o 6352  df-fv 6353  df-pmtr 18572
This theorem is referenced by:  pmtrfv  18582  pmtrf  18585  cycpm2tr  30803  trsp2cyc  30807
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