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Theorem pmtrval 19449
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 19448 . . . 4 (𝐷𝑉𝑇 = (𝑝 ∈ {𝑦 ∈ 𝒫 𝐷𝑦 ≈ 2o} ↦ (𝑧𝐷 ↦ if(𝑧𝑝, (𝑝 ∖ {𝑧}), 𝑧))))
32fveq1d 6903 . . 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 2726 . . 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 2815 . . . . 5 (𝑝 = 𝑃 → (𝑧𝑝𝑧𝑃))
7 difeq1 4114 . . . . . 6 (𝑝 = 𝑃 → (𝑝 ∖ {𝑧}) = (𝑃 ∖ {𝑧}))
87unieqd 4926 . . . . 5 (𝑝 = 𝑃 (𝑝 ∖ {𝑧}) = (𝑃 ∖ {𝑧}))
96, 8ifbieq1d 4557 . . . 4 (𝑝 = 𝑃 → if(𝑧𝑝, (𝑝 ∖ {𝑧}), 𝑧) = if(𝑧𝑃, (𝑃 ∖ {𝑧}), 𝑧))
109mpteq2dv 5255 . . 3 (𝑝 = 𝑃 → (𝑧𝐷 ↦ if(𝑧𝑝, (𝑝 ∖ {𝑧}), 𝑧)) = (𝑧𝐷 ↦ if(𝑧𝑃, (𝑃 ∖ {𝑧}), 𝑧)))
11 breq1 5156 . . . 4 (𝑦 = 𝑃 → (𝑦 ≈ 2o𝑃 ≈ 2o))
12 elpw2g 5351 . . . . . 6 (𝐷𝑉 → (𝑃 ∈ 𝒫 𝐷𝑃𝐷))
1312biimpar 476 . . . . 5 ((𝐷𝑉𝑃𝐷) → 𝑃 ∈ 𝒫 𝐷)
14133adant3 1129 . . . 4 ((𝐷𝑉𝑃𝐷𝑃 ≈ 2o) → 𝑃 ∈ 𝒫 𝐷)
15 simp3 1135 . . . 4 ((𝐷𝑉𝑃𝐷𝑃 ≈ 2o) → 𝑃 ≈ 2o)
1611, 14, 15elrabd 3683 . . 3 ((𝐷𝑉𝑃𝐷𝑃 ≈ 2o) → 𝑃 ∈ {𝑦 ∈ 𝒫 𝐷𝑦 ≈ 2o})
17 mptexg 7238 . . . 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 7032 . 2 ((𝐷𝑉𝑃𝐷𝑃 ≈ 2o) → ((𝑝 ∈ {𝑦 ∈ 𝒫 𝐷𝑦 ≈ 2o} ↦ (𝑧𝐷 ↦ if(𝑧𝑝, (𝑝 ∖ {𝑧}), 𝑧)))‘𝑃) = (𝑧𝐷 ↦ if(𝑧𝑃, (𝑃 ∖ {𝑧}), 𝑧)))
204, 19eqtrd 2766 1 ((𝐷𝑉𝑃𝐷𝑃 ≈ 2o) → (𝑇𝑃) = (𝑧𝐷 ↦ if(𝑧𝑃, (𝑃 ∖ {𝑧}), 𝑧)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1084   = wceq 1534  wcel 2099  {crab 3419  Vcvv 3462  cdif 3944  wss 3947  ifcif 4533  𝒫 cpw 4607  {csn 4633   cuni 4913   class class class wbr 5153  cmpt 5236  cfv 6554  2oc2o 8490  cen 8971  pmTrspcpmtr 19439
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-rep 5290  ax-sep 5304  ax-nul 5311  ax-pow 5369  ax-pr 5433
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ne 2931  df-ral 3052  df-rex 3061  df-reu 3365  df-rab 3420  df-v 3464  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4326  df-if 4534  df-pw 4609  df-sn 4634  df-pr 4636  df-op 4640  df-uni 4914  df-iun 5003  df-br 5154  df-opab 5216  df-mpt 5237  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-iota 6506  df-fun 6556  df-fn 6557  df-f 6558  df-f1 6559  df-fo 6560  df-f1o 6561  df-fv 6562  df-pmtr 19440
This theorem is referenced by:  pmtrfv  19450  pmtrf  19453  cycpm2tr  32997  trsp2cyc  33001
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