Theorem pmtrval 19493

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 19492	. . . 4 ⊢ (𝐷 ∈ 𝑉 → 𝑇 = (𝑝 ∈ {𝑦 ∈ 𝒫 𝐷 ∣ 𝑦 ≈ 2o} ↦ (𝑧 ∈ 𝐷 ↦ if(𝑧 ∈ 𝑝, ∪ (𝑝 ∖ {𝑧}), 𝑧))))
3	2	fveq1d 6922	. . 3 ⊢ (𝐷 ∈ 𝑉 → (𝑇‘𝑃) = ((𝑝 ∈ {𝑦 ∈ 𝒫 𝐷 ∣ 𝑦 ≈ 2o} ↦ (𝑧 ∈ 𝐷 ↦ if(𝑧 ∈ 𝑝, ∪ (𝑝 ∖ {𝑧}), 𝑧)))‘𝑃))
4	3	3ad2ant1 1133	. 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 2740	. . 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 2833	. . . . 5 ⊢ (𝑝 = 𝑃 → (𝑧 ∈ 𝑝 ↔ 𝑧 ∈ 𝑃))
7		difeq1 4142	. . . . . 6 ⊢ (𝑝 = 𝑃 → (𝑝 ∖ {𝑧}) = (𝑃 ∖ {𝑧}))
8	7	unieqd 4944	. . . . 5 ⊢ (𝑝 = 𝑃 → ∪ (𝑝 ∖ {𝑧}) = ∪ (𝑃 ∖ {𝑧}))
9	6, 8	ifbieq1d 4572	. . . 4 ⊢ (𝑝 = 𝑃 → if(𝑧 ∈ 𝑝, ∪ (𝑝 ∖ {𝑧}), 𝑧) = if(𝑧 ∈ 𝑃, ∪ (𝑃 ∖ {𝑧}), 𝑧))
10	9	mpteq2dv 5268	. . 3 ⊢ (𝑝 = 𝑃 → (𝑧 ∈ 𝐷 ↦ if(𝑧 ∈ 𝑝, ∪ (𝑝 ∖ {𝑧}), 𝑧)) = (𝑧 ∈ 𝐷 ↦ if(𝑧 ∈ 𝑃, ∪ (𝑃 ∖ {𝑧}), 𝑧)))
11		breq1 5169	. . . 4 ⊢ (𝑦 = 𝑃 → (𝑦 ≈ 2o ↔ 𝑃 ≈ 2o))
12		elpw2g 5351	. . . . . 6 ⊢ (𝐷 ∈ 𝑉 → (𝑃 ∈ 𝒫 𝐷 ↔ 𝑃 ⊆ 𝐷))
13	12	biimpar 477	. . . . 5 ⊢ ((𝐷 ∈ 𝑉 ∧ 𝑃 ⊆ 𝐷) → 𝑃 ∈ 𝒫 𝐷)
14	13	3adant3 1132	. . . 4 ⊢ ((𝐷 ∈ 𝑉 ∧ 𝑃 ⊆ 𝐷 ∧ 𝑃 ≈ 2o) → 𝑃 ∈ 𝒫 𝐷)
15		simp3 1138	. . . 4 ⊢ ((𝐷 ∈ 𝑉 ∧ 𝑃 ⊆ 𝐷 ∧ 𝑃 ≈ 2o) → 𝑃 ≈ 2o)
16	11, 14, 15	elrabd 3710	. . 3 ⊢ ((𝐷 ∈ 𝑉 ∧ 𝑃 ⊆ 𝐷 ∧ 𝑃 ≈ 2o) → 𝑃 ∈ {𝑦 ∈ 𝒫 𝐷 ∣ 𝑦 ≈ 2o})
17		mptexg 7258	. . . 4 ⊢ (𝐷 ∈ 𝑉 → (𝑧 ∈ 𝐷 ↦ if(𝑧 ∈ 𝑃, ∪ (𝑃 ∖ {𝑧}), 𝑧)) ∈ V)
18	17	3ad2ant1 1133	. . 3 ⊢ ((𝐷 ∈ 𝑉 ∧ 𝑃 ⊆ 𝐷 ∧ 𝑃 ≈ 2o) → (𝑧 ∈ 𝐷 ↦ if(𝑧 ∈ 𝑃, ∪ (𝑃 ∖ {𝑧}), 𝑧)) ∈ V)
19	5, 10, 16, 18	fvmptd3 7052	. 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 2780	1 ⊢ ((𝐷 ∈ 𝑉 ∧ 𝑃 ⊆ 𝐷 ∧ 𝑃 ≈ 2o) → (𝑇‘𝑃) = (𝑧 ∈ 𝐷 ↦ if(𝑧 ∈ 𝑃, ∪ (𝑃 ∖ {𝑧}), 𝑧)))

Syntax hints:   → wi 4   ∧ w3a 1087   = wceq 1537   ∈ wcel 2108  {crab 3443  Vcvv 3488   ∖ cdif 3973   ⊆ wss 3976  ifcif 4548  𝒫 cpw 4622  {csn 4648  ∪ cuni 4931   class class class wbr 5166   ↦ cmpt 5249  ‘cfv 6573  2_oc2o 8516   ≈ cen 9000  pmTrspcpmtr 19483

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-rep 5303  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-iun 5017  df-br 5167  df-opab 5229  df-mpt 5250  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-pmtr 19484

This theorem is referenced by:  pmtrfv  19494  pmtrf  19497  cycpm2tr  33112  trsp2cyc  33116