Theorem pmtrval 19463

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	⊢ ((𝐷 ∈ 𝑉 ∧ 𝑃 ⊆ 𝐷 ∧ 𝑃 ≈ 2o) → (𝑇‘𝑃) = (𝑧 ∈ 𝐷 ↦ if(𝑧 ∈ 𝑃, ∪ (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.

Step	Hyp	Ref	Expression
1		pmtrfval.t	. . . . 5 ⊢ 𝑇 = (pmTrsp‘𝐷)
2	1	pmtrfval 19462	. . . 4 ⊢ (𝐷 ∈ 𝑉 → 𝑇 = (𝑝 ∈ {𝑦 ∈ 𝒫 𝐷 ∣ 𝑦 ≈ 2o} ↦ (𝑧 ∈ 𝐷 ↦ if(𝑧 ∈ 𝑝, ∪ (𝑝 ∖ {𝑧}), 𝑧))))
3	2	fveq1d 6854	. . 3 ⊢ (𝐷 ∈ 𝑉 → (𝑇‘𝑃) = ((𝑝 ∈ {𝑦 ∈ 𝒫 𝐷 ∣ 𝑦 ≈ 2o} ↦ (𝑧 ∈ 𝐷 ↦ if(𝑧 ∈ 𝑝, ∪ (𝑝 ∖ {𝑧}), 𝑧)))‘𝑃))
4	3	3ad2ant1 1142	. 2 ⊢ ((𝐷 ∈ 𝑉 ∧ 𝑃 ⊆ 𝐷 ∧ 𝑃 ≈ 2o) → (𝑇‘𝑃) = ((𝑝 ∈ {𝑦 ∈ 𝒫 𝐷 ∣ 𝑦 ≈ 2o} ↦ (𝑧 ∈ 𝐷 ↦ if(𝑧 ∈ 𝑝, ∪ (𝑝 ∖ {𝑧}), 𝑧)))‘𝑃))
5		eqid 2752	. . 3 ⊢ (𝑝 ∈ {𝑦 ∈ 𝒫 𝐷 ∣ 𝑦 ≈ 2o} ↦ (𝑧 ∈ 𝐷 ↦ if(𝑧 ∈ 𝑝, ∪ (𝑝 ∖ {𝑧}), 𝑧))) = (𝑝 ∈ {𝑦 ∈ 𝒫 𝐷 ∣ 𝑦 ≈ 2o} ↦ (𝑧 ∈ 𝐷 ↦ if(𝑧 ∈ 𝑝, ∪ (𝑝 ∖ {𝑧}), 𝑧)))
6		eleq2 2841	. . . . 5 ⊢ (𝑝 = 𝑃 → (𝑧 ∈ 𝑝 ↔ 𝑧 ∈ 𝑃))
7		difeq1 4064	. . . . . 6 ⊢ (𝑝 = 𝑃 → (𝑝 ∖ {𝑧}) = (𝑃 ∖ {𝑧}))
8	7	unieqd 4868	. . . . 5 ⊢ (𝑝 = 𝑃 → ∪ (𝑝 ∖ {𝑧}) = ∪ (𝑃 ∖ {𝑧}))
9	6, 8	ifbieq1d 4495	. . . 4 ⊢ (𝑝 = 𝑃 → if(𝑧 ∈ 𝑝, ∪ (𝑝 ∖ {𝑧}), 𝑧) = if(𝑧 ∈ 𝑃, ∪ (𝑃 ∖ {𝑧}), 𝑧))
10	9	mpteq2dv 5184	. . 3 ⊢ (𝑝 = 𝑃 → (𝑧 ∈ 𝐷 ↦ if(𝑧 ∈ 𝑝, ∪ (𝑝 ∖ {𝑧}), 𝑧)) = (𝑧 ∈ 𝐷 ↦ if(𝑧 ∈ 𝑃, ∪ (𝑃 ∖ {𝑧}), 𝑧)))
11		breq1 5093	. . . 4 ⊢ (𝑦 = 𝑃 → (𝑦 ≈ 2o ↔ 𝑃 ≈ 2o))
12		elpw2g 5279	. . . . . 6 ⊢ (𝐷 ∈ 𝑉 → (𝑃 ∈ 𝒫 𝐷 ↔ 𝑃 ⊆ 𝐷))
13	12	biimpar 480	. . . . 5 ⊢ ((𝐷 ∈ 𝑉 ∧ 𝑃 ⊆ 𝐷) → 𝑃 ∈ 𝒫 𝐷)
14	13	3adant3 1141	. . . 4 ⊢ ((𝐷 ∈ 𝑉 ∧ 𝑃 ⊆ 𝐷 ∧ 𝑃 ≈ 2o) → 𝑃 ∈ 𝒫 𝐷)
15		simp3 1147	. . . 4 ⊢ ((𝐷 ∈ 𝑉 ∧ 𝑃 ⊆ 𝐷 ∧ 𝑃 ≈ 2o) → 𝑃 ≈ 2o)
16	11, 14, 15	elrabd 3643	. . 3 ⊢ ((𝐷 ∈ 𝑉 ∧ 𝑃 ⊆ 𝐷 ∧ 𝑃 ≈ 2o) → 𝑃 ∈ {𝑦 ∈ 𝒫 𝐷 ∣ 𝑦 ≈ 2o})
17		mptexg 7190	. . . 4 ⊢ (𝐷 ∈ 𝑉 → (𝑧 ∈ 𝐷 ↦ if(𝑧 ∈ 𝑃, ∪ (𝑃 ∖ {𝑧}), 𝑧)) ∈ V)
18	17	3ad2ant1 1142	. . 3 ⊢ ((𝐷 ∈ 𝑉 ∧ 𝑃 ⊆ 𝐷 ∧ 𝑃 ≈ 2o) → (𝑧 ∈ 𝐷 ↦ if(𝑧 ∈ 𝑃, ∪ (𝑃 ∖ {𝑧}), 𝑧)) ∈ V)
19	5, 10, 16, 18	fvmptd3 6984	. 2 ⊢ ((𝐷 ∈ 𝑉 ∧ 𝑃 ⊆ 𝐷 ∧ 𝑃 ≈ 2o) → ((𝑝 ∈ {𝑦 ∈ 𝒫 𝐷 ∣ 𝑦 ≈ 2o} ↦ (𝑧 ∈ 𝐷 ↦ if(𝑧 ∈ 𝑝, ∪ (𝑝 ∖ {𝑧}), 𝑧)))‘𝑃) = (𝑧 ∈ 𝐷 ↦ if(𝑧 ∈ 𝑃, ∪ (𝑃 ∖ {𝑧}), 𝑧)))
20	4, 19	eqtrd 2787	1 ⊢ ((𝐷 ∈ 𝑉 ∧ 𝑃 ⊆ 𝐷 ∧ 𝑃 ≈ 2o) → (𝑇‘𝑃) = (𝑧 ∈ 𝐷 ↦ if(𝑧 ∈ 𝑃, ∪ (𝑃 ∖ {𝑧}), 𝑧)))

Syntax hints:   → wi 4   ∧ w3a 1095   = wceq 1550   ∈ wcel 2132  {crab 3404  Vcvv 3444   ∖ cdif 3892   ⊆ wss 3895  ifcif 4470  𝒫 cpw 4545  {csn 4572  ∪ cuni 4855   class class class wbr 5090   ↦ cmpt 5171  ‘cfv 6506  2_oc2o 8415   ≈ cen 8909  pmTrspcpmtr 19453

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1805  ax-4 1819  ax-5 1920  ax-6 1977  ax-7 2018  ax-8 2134  ax-9 2142  ax-10 2165  ax-11 2181  ax-12 2202  ax-ext 2724  ax-rep 5217  ax-sep 5236  ax-nul 5246  ax-pow 5312  ax-pr 5380

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 857  df-3an 1097  df-tru 1553  df-fal 1563  df-ex 1790  df-nf 1794  df-sb 2081  df-mo 2556  df-eu 2586  df-clab 2731  df-cleq 2744  df-clel 2827  df-nfc 2901  df-ne 2948  df-ral 3067  df-rex 3077  df-reu 3358  df-rab 3405  df-v 3446  df-sbc 3736  df-csb 3844  df-dif 3898  df-un 3900  df-in 3902  df-ss 3912  df-nul 4277  df-if 4471  df-pw 4547  df-sn 4573  df-pr 4575  df-op 4579  df-uni 4856  df-iun 4941  df-br 5091  df-opab 5153  df-mpt 5172  df-id 5531  df-xp 5642  df-rel 5643  df-cnv 5644  df-co 5645  df-dm 5646  df-rn 5647  df-res 5648  df-ima 5649  df-iota 6462  df-fun 6508  df-fn 6509  df-f 6510  df-f1 6511  df-fo 6512  df-f1o 6513  df-fv 6514  df-pmtr 19454

This theorem is referenced by:  pmtrfv  19464  pmtrf  19467  cycpm2tr  33249  trsp2cyc  33253