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Theorem pmtrval 17917
 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 17916 . . . 4 (𝐷𝑉𝑇 = (𝑝 ∈ {𝑦 ∈ 𝒫 𝐷𝑦 ≈ 2𝑜} ↦ (𝑧𝐷 ↦ if(𝑧𝑝, (𝑝 ∖ {𝑧}), 𝑧))))
32fveq1d 6231 . . 3 (𝐷𝑉 → (𝑇𝑃) = ((𝑝 ∈ {𝑦 ∈ 𝒫 𝐷𝑦 ≈ 2𝑜} ↦ (𝑧𝐷 ↦ if(𝑧𝑝, (𝑝 ∖ {𝑧}), 𝑧)))‘𝑃))
433ad2ant1 1102 . 2 ((𝐷𝑉𝑃𝐷𝑃 ≈ 2𝑜) → (𝑇𝑃) = ((𝑝 ∈ {𝑦 ∈ 𝒫 𝐷𝑦 ≈ 2𝑜} ↦ (𝑧𝐷 ↦ if(𝑧𝑝, (𝑝 ∖ {𝑧}), 𝑧)))‘𝑃))
5 elpw2g 4857 . . . . . 6 (𝐷𝑉 → (𝑃 ∈ 𝒫 𝐷𝑃𝐷))
65biimpar 501 . . . . 5 ((𝐷𝑉𝑃𝐷) → 𝑃 ∈ 𝒫 𝐷)
763adant3 1101 . . . 4 ((𝐷𝑉𝑃𝐷𝑃 ≈ 2𝑜) → 𝑃 ∈ 𝒫 𝐷)
8 simp3 1083 . . . 4 ((𝐷𝑉𝑃𝐷𝑃 ≈ 2𝑜) → 𝑃 ≈ 2𝑜)
9 breq1 4688 . . . . 5 (𝑦 = 𝑃 → (𝑦 ≈ 2𝑜𝑃 ≈ 2𝑜))
109elrab 3396 . . . 4 (𝑃 ∈ {𝑦 ∈ 𝒫 𝐷𝑦 ≈ 2𝑜} ↔ (𝑃 ∈ 𝒫 𝐷𝑃 ≈ 2𝑜))
117, 8, 10sylanbrc 699 . . 3 ((𝐷𝑉𝑃𝐷𝑃 ≈ 2𝑜) → 𝑃 ∈ {𝑦 ∈ 𝒫 𝐷𝑦 ≈ 2𝑜})
12 mptexg 6525 . . . 4 (𝐷𝑉 → (𝑧𝐷 ↦ if(𝑧𝑃, (𝑃 ∖ {𝑧}), 𝑧)) ∈ V)
13123ad2ant1 1102 . . 3 ((𝐷𝑉𝑃𝐷𝑃 ≈ 2𝑜) → (𝑧𝐷 ↦ if(𝑧𝑃, (𝑃 ∖ {𝑧}), 𝑧)) ∈ V)
14 eleq2 2719 . . . . . 6 (𝑝 = 𝑃 → (𝑧𝑝𝑧𝑃))
15 difeq1 3754 . . . . . . 7 (𝑝 = 𝑃 → (𝑝 ∖ {𝑧}) = (𝑃 ∖ {𝑧}))
1615unieqd 4478 . . . . . 6 (𝑝 = 𝑃 (𝑝 ∖ {𝑧}) = (𝑃 ∖ {𝑧}))
1714, 16ifbieq1d 4142 . . . . 5 (𝑝 = 𝑃 → if(𝑧𝑝, (𝑝 ∖ {𝑧}), 𝑧) = if(𝑧𝑃, (𝑃 ∖ {𝑧}), 𝑧))
1817mpteq2dv 4778 . . . 4 (𝑝 = 𝑃 → (𝑧𝐷 ↦ if(𝑧𝑝, (𝑝 ∖ {𝑧}), 𝑧)) = (𝑧𝐷 ↦ if(𝑧𝑃, (𝑃 ∖ {𝑧}), 𝑧)))
19 eqid 2651 . . . 4 (𝑝 ∈ {𝑦 ∈ 𝒫 𝐷𝑦 ≈ 2𝑜} ↦ (𝑧𝐷 ↦ if(𝑧𝑝, (𝑝 ∖ {𝑧}), 𝑧))) = (𝑝 ∈ {𝑦 ∈ 𝒫 𝐷𝑦 ≈ 2𝑜} ↦ (𝑧𝐷 ↦ if(𝑧𝑝, (𝑝 ∖ {𝑧}), 𝑧)))
2018, 19fvmptg 6319 . . 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)))
2111, 13, 20syl2anc 694 . 2 ((𝐷𝑉𝑃𝐷𝑃 ≈ 2𝑜) → ((𝑝 ∈ {𝑦 ∈ 𝒫 𝐷𝑦 ≈ 2𝑜} ↦ (𝑧𝐷 ↦ if(𝑧𝑝, (𝑝 ∖ {𝑧}), 𝑧)))‘𝑃) = (𝑧𝐷 ↦ if(𝑧𝑃, (𝑃 ∖ {𝑧}), 𝑧)))
224, 21eqtrd 2685 1 ((𝐷𝑉𝑃𝐷𝑃 ≈ 2𝑜) → (𝑇𝑃) = (𝑧𝐷 ↦ if(𝑧𝑃, (𝑃 ∖ {𝑧}), 𝑧)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ w3a 1054   = wceq 1523   ∈ wcel 2030  {crab 2945  Vcvv 3231   ∖ cdif 3604   ⊆ wss 3607  ifcif 4119  𝒫 cpw 4191  {csn 4210  ∪ cuni 4468   class class class wbr 4685   ↦ cmpt 4762  ‘cfv 5926  2𝑜c2o 7599   ≈ cen 7994  pmTrspcpmtr 17907 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-pmtr 17908 This theorem is referenced by:  pmtrfv  17918  pmtrf  17921
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