MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  pmtrval Structured version   Visualization version   GIF version

Theorem pmtrval 18091
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 18090 . . . 4 (𝐷𝑉𝑇 = (𝑝 ∈ {𝑦 ∈ 𝒫 𝐷𝑦 ≈ 2𝑜} ↦ (𝑧𝐷 ↦ if(𝑧𝑝, (𝑝 ∖ {𝑧}), 𝑧))))
32fveq1d 6355 . . 3 (𝐷𝑉 → (𝑇𝑃) = ((𝑝 ∈ {𝑦 ∈ 𝒫 𝐷𝑦 ≈ 2𝑜} ↦ (𝑧𝐷 ↦ if(𝑧𝑝, (𝑝 ∖ {𝑧}), 𝑧)))‘𝑃))
433ad2ant1 1128 . 2 ((𝐷𝑉𝑃𝐷𝑃 ≈ 2𝑜) → (𝑇𝑃) = ((𝑝 ∈ {𝑦 ∈ 𝒫 𝐷𝑦 ≈ 2𝑜} ↦ (𝑧𝐷 ↦ if(𝑧𝑝, (𝑝 ∖ {𝑧}), 𝑧)))‘𝑃))
5 elpw2g 4976 . . . . . 6 (𝐷𝑉 → (𝑃 ∈ 𝒫 𝐷𝑃𝐷))
65biimpar 503 . . . . 5 ((𝐷𝑉𝑃𝐷) → 𝑃 ∈ 𝒫 𝐷)
763adant3 1127 . . . 4 ((𝐷𝑉𝑃𝐷𝑃 ≈ 2𝑜) → 𝑃 ∈ 𝒫 𝐷)
8 simp3 1133 . . . 4 ((𝐷𝑉𝑃𝐷𝑃 ≈ 2𝑜) → 𝑃 ≈ 2𝑜)
9 breq1 4807 . . . . 5 (𝑦 = 𝑃 → (𝑦 ≈ 2𝑜𝑃 ≈ 2𝑜))
109elrab 3504 . . . 4 (𝑃 ∈ {𝑦 ∈ 𝒫 𝐷𝑦 ≈ 2𝑜} ↔ (𝑃 ∈ 𝒫 𝐷𝑃 ≈ 2𝑜))
117, 8, 10sylanbrc 701 . . 3 ((𝐷𝑉𝑃𝐷𝑃 ≈ 2𝑜) → 𝑃 ∈ {𝑦 ∈ 𝒫 𝐷𝑦 ≈ 2𝑜})
12 mptexg 6649 . . . 4 (𝐷𝑉 → (𝑧𝐷 ↦ if(𝑧𝑃, (𝑃 ∖ {𝑧}), 𝑧)) ∈ V)
13123ad2ant1 1128 . . 3 ((𝐷𝑉𝑃𝐷𝑃 ≈ 2𝑜) → (𝑧𝐷 ↦ if(𝑧𝑃, (𝑃 ∖ {𝑧}), 𝑧)) ∈ V)
14 eleq2 2828 . . . . . 6 (𝑝 = 𝑃 → (𝑧𝑝𝑧𝑃))
15 difeq1 3864 . . . . . . 7 (𝑝 = 𝑃 → (𝑝 ∖ {𝑧}) = (𝑃 ∖ {𝑧}))
1615unieqd 4598 . . . . . 6 (𝑝 = 𝑃 (𝑝 ∖ {𝑧}) = (𝑃 ∖ {𝑧}))
1714, 16ifbieq1d 4253 . . . . 5 (𝑝 = 𝑃 → if(𝑧𝑝, (𝑝 ∖ {𝑧}), 𝑧) = if(𝑧𝑃, (𝑃 ∖ {𝑧}), 𝑧))
1817mpteq2dv 4897 . . . 4 (𝑝 = 𝑃 → (𝑧𝐷 ↦ if(𝑧𝑝, (𝑝 ∖ {𝑧}), 𝑧)) = (𝑧𝐷 ↦ if(𝑧𝑃, (𝑃 ∖ {𝑧}), 𝑧)))
19 eqid 2760 . . . 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 6443 . . 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 696 . 2 ((𝐷𝑉𝑃𝐷𝑃 ≈ 2𝑜) → ((𝑝 ∈ {𝑦 ∈ 𝒫 𝐷𝑦 ≈ 2𝑜} ↦ (𝑧𝐷 ↦ if(𝑧𝑝, (𝑝 ∖ {𝑧}), 𝑧)))‘𝑃) = (𝑧𝐷 ↦ if(𝑧𝑃, (𝑃 ∖ {𝑧}), 𝑧)))
224, 21eqtrd 2794 1 ((𝐷𝑉𝑃𝐷𝑃 ≈ 2𝑜) → (𝑇𝑃) = (𝑧𝐷 ↦ if(𝑧𝑃, (𝑃 ∖ {𝑧}), 𝑧)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1072   = wceq 1632  wcel 2139  {crab 3054  Vcvv 3340  cdif 3712  wss 3715  ifcif 4230  𝒫 cpw 4302  {csn 4321   cuni 4588   class class class wbr 4804  cmpt 4881  cfv 6049  2𝑜c2o 7724  cen 8120  pmTrspcpmtr 18081
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-reu 3057  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-pmtr 18082
This theorem is referenced by:  pmtrfv  18092  pmtrf  18095
  Copyright terms: Public domain W3C validator