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Theorem pmtrval 18265
 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 18264 . . . 4 (𝐷𝑉𝑇 = (𝑝 ∈ {𝑦 ∈ 𝒫 𝐷𝑦 ≈ 2o} ↦ (𝑧𝐷 ↦ if(𝑧𝑝, (𝑝 ∖ {𝑧}), 𝑧))))
32fveq1d 6450 . . 3 (𝐷𝑉 → (𝑇𝑃) = ((𝑝 ∈ {𝑦 ∈ 𝒫 𝐷𝑦 ≈ 2o} ↦ (𝑧𝐷 ↦ if(𝑧𝑝, (𝑝 ∖ {𝑧}), 𝑧)))‘𝑃))
433ad2ant1 1124 . 2 ((𝐷𝑉𝑃𝐷𝑃 ≈ 2o) → (𝑇𝑃) = ((𝑝 ∈ {𝑦 ∈ 𝒫 𝐷𝑦 ≈ 2o} ↦ (𝑧𝐷 ↦ if(𝑧𝑝, (𝑝 ∖ {𝑧}), 𝑧)))‘𝑃))
5 breq1 4891 . . . 4 (𝑦 = 𝑃 → (𝑦 ≈ 2o𝑃 ≈ 2o))
6 elpw2g 5063 . . . . . 6 (𝐷𝑉 → (𝑃 ∈ 𝒫 𝐷𝑃𝐷))
76biimpar 471 . . . . 5 ((𝐷𝑉𝑃𝐷) → 𝑃 ∈ 𝒫 𝐷)
873adant3 1123 . . . 4 ((𝐷𝑉𝑃𝐷𝑃 ≈ 2o) → 𝑃 ∈ 𝒫 𝐷)
9 simp3 1129 . . . 4 ((𝐷𝑉𝑃𝐷𝑃 ≈ 2o) → 𝑃 ≈ 2o)
105, 8, 9elrabd 3575 . . 3 ((𝐷𝑉𝑃𝐷𝑃 ≈ 2o) → 𝑃 ∈ {𝑦 ∈ 𝒫 𝐷𝑦 ≈ 2o})
11 mptexg 6758 . . . 4 (𝐷𝑉 → (𝑧𝐷 ↦ if(𝑧𝑃, (𝑃 ∖ {𝑧}), 𝑧)) ∈ V)
12113ad2ant1 1124 . . 3 ((𝐷𝑉𝑃𝐷𝑃 ≈ 2o) → (𝑧𝐷 ↦ if(𝑧𝑃, (𝑃 ∖ {𝑧}), 𝑧)) ∈ V)
13 eleq2 2848 . . . . . 6 (𝑝 = 𝑃 → (𝑧𝑝𝑧𝑃))
14 difeq1 3944 . . . . . . 7 (𝑝 = 𝑃 → (𝑝 ∖ {𝑧}) = (𝑃 ∖ {𝑧}))
1514unieqd 4683 . . . . . 6 (𝑝 = 𝑃 (𝑝 ∖ {𝑧}) = (𝑃 ∖ {𝑧}))
1613, 15ifbieq1d 4330 . . . . 5 (𝑝 = 𝑃 → if(𝑧𝑝, (𝑝 ∖ {𝑧}), 𝑧) = if(𝑧𝑃, (𝑃 ∖ {𝑧}), 𝑧))
1716mpteq2dv 4982 . . . 4 (𝑝 = 𝑃 → (𝑧𝐷 ↦ if(𝑧𝑝, (𝑝 ∖ {𝑧}), 𝑧)) = (𝑧𝐷 ↦ if(𝑧𝑃, (𝑃 ∖ {𝑧}), 𝑧)))
18 eqid 2778 . . . 4 (𝑝 ∈ {𝑦 ∈ 𝒫 𝐷𝑦 ≈ 2o} ↦ (𝑧𝐷 ↦ if(𝑧𝑝, (𝑝 ∖ {𝑧}), 𝑧))) = (𝑝 ∈ {𝑦 ∈ 𝒫 𝐷𝑦 ≈ 2o} ↦ (𝑧𝐷 ↦ if(𝑧𝑝, (𝑝 ∖ {𝑧}), 𝑧)))
1917, 18fvmptg 6542 . . 3 ((𝑃 ∈ {𝑦 ∈ 𝒫 𝐷𝑦 ≈ 2o} ∧ (𝑧𝐷 ↦ if(𝑧𝑃, (𝑃 ∖ {𝑧}), 𝑧)) ∈ V) → ((𝑝 ∈ {𝑦 ∈ 𝒫 𝐷𝑦 ≈ 2o} ↦ (𝑧𝐷 ↦ if(𝑧𝑝, (𝑝 ∖ {𝑧}), 𝑧)))‘𝑃) = (𝑧𝐷 ↦ if(𝑧𝑃, (𝑃 ∖ {𝑧}), 𝑧)))
2010, 12, 19syl2anc 579 . 2 ((𝐷𝑉𝑃𝐷𝑃 ≈ 2o) → ((𝑝 ∈ {𝑦 ∈ 𝒫 𝐷𝑦 ≈ 2o} ↦ (𝑧𝐷 ↦ if(𝑧𝑝, (𝑝 ∖ {𝑧}), 𝑧)))‘𝑃) = (𝑧𝐷 ↦ if(𝑧𝑃, (𝑃 ∖ {𝑧}), 𝑧)))
214, 20eqtrd 2814 1 ((𝐷𝑉𝑃𝐷𝑃 ≈ 2o) → (𝑇𝑃) = (𝑧𝐷 ↦ if(𝑧𝑃, (𝑃 ∖ {𝑧}), 𝑧)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ w3a 1071   = wceq 1601   ∈ wcel 2107  {crab 3094  Vcvv 3398   ∖ cdif 3789   ⊆ wss 3792  ifcif 4307  𝒫 cpw 4379  {csn 4398  ∪ cuni 4673   class class class wbr 4888   ↦ cmpt 4967  ‘cfv 6137  2oc2o 7839   ≈ cen 8240  pmTrspcpmtr 18255 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-rep 5008  ax-sep 5019  ax-nul 5027  ax-pow 5079  ax-pr 5140 This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ne 2970  df-ral 3095  df-rex 3096  df-reu 3097  df-rab 3099  df-v 3400  df-sbc 3653  df-csb 3752  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-nul 4142  df-if 4308  df-pw 4381  df-sn 4399  df-pr 4401  df-op 4405  df-uni 4674  df-iun 4757  df-br 4889  df-opab 4951  df-mpt 4968  df-id 5263  df-xp 5363  df-rel 5364  df-cnv 5365  df-co 5366  df-dm 5367  df-rn 5368  df-res 5369  df-ima 5370  df-iota 6101  df-fun 6139  df-fn 6140  df-f 6141  df-f1 6142  df-fo 6143  df-f1o 6144  df-fv 6145  df-pmtr 18256 This theorem is referenced by:  pmtrfv  18266  pmtrf  18269
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