Theorem pmtrval 17871

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 17870	. . . 4 ⊢ (𝐷 ∈ 𝑉 → 𝑇 = (𝑝 ∈ {𝑦 ∈ 𝒫 𝐷 ∣ 𝑦 ≈ 2𝑜} ↦ (𝑧 ∈ 𝐷 ↦ if(𝑧 ∈ 𝑝, ∪ (𝑝 ∖ {𝑧}), 𝑧))))
3	2	fveq1d 6193	. . 3 ⊢ (𝐷 ∈ 𝑉 → (𝑇‘𝑃) = ((𝑝 ∈ {𝑦 ∈ 𝒫 𝐷 ∣ 𝑦 ≈ 2𝑜} ↦ (𝑧 ∈ 𝐷 ↦ if(𝑧 ∈ 𝑝, ∪ (𝑝 ∖ {𝑧}), 𝑧)))‘𝑃))
4	3	3ad2ant1 1082	. 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		elpw2g 4827	. . . . . 6 ⊢ (𝐷 ∈ 𝑉 → (𝑃 ∈ 𝒫 𝐷 ↔ 𝑃 ⊆ 𝐷))
6	5	biimpar 502	. . . . 5 ⊢ ((𝐷 ∈ 𝑉 ∧ 𝑃 ⊆ 𝐷) → 𝑃 ∈ 𝒫 𝐷)
7	6	3adant3 1081	. . . 4 ⊢ ((𝐷 ∈ 𝑉 ∧ 𝑃 ⊆ 𝐷 ∧ 𝑃 ≈ 2𝑜) → 𝑃 ∈ 𝒫 𝐷)
8		simp3 1063	. . . 4 ⊢ ((𝐷 ∈ 𝑉 ∧ 𝑃 ⊆ 𝐷 ∧ 𝑃 ≈ 2𝑜) → 𝑃 ≈ 2𝑜)
9		breq1 4656	. . . . 5 ⊢ (𝑦 = 𝑃 → (𝑦 ≈ 2𝑜 ↔ 𝑃 ≈ 2𝑜))
10	9	elrab 3363	. . . 4 ⊢ (𝑃 ∈ {𝑦 ∈ 𝒫 𝐷 ∣ 𝑦 ≈ 2𝑜} ↔ (𝑃 ∈ 𝒫 𝐷 ∧ 𝑃 ≈ 2𝑜))
11	7, 8, 10	sylanbrc 698	. . 3 ⊢ ((𝐷 ∈ 𝑉 ∧ 𝑃 ⊆ 𝐷 ∧ 𝑃 ≈ 2𝑜) → 𝑃 ∈ {𝑦 ∈ 𝒫 𝐷 ∣ 𝑦 ≈ 2𝑜})
12		mptexg 6484	. . . 4 ⊢ (𝐷 ∈ 𝑉 → (𝑧 ∈ 𝐷 ↦ if(𝑧 ∈ 𝑃, ∪ (𝑃 ∖ {𝑧}), 𝑧)) ∈ V)
13	12	3ad2ant1 1082	. . 3 ⊢ ((𝐷 ∈ 𝑉 ∧ 𝑃 ⊆ 𝐷 ∧ 𝑃 ≈ 2𝑜) → (𝑧 ∈ 𝐷 ↦ if(𝑧 ∈ 𝑃, ∪ (𝑃 ∖ {𝑧}), 𝑧)) ∈ V)
14		eleq2 2690	. . . . . 6 ⊢ (𝑝 = 𝑃 → (𝑧 ∈ 𝑝 ↔ 𝑧 ∈ 𝑃))
15		difeq1 3721	. . . . . . 7 ⊢ (𝑝 = 𝑃 → (𝑝 ∖ {𝑧}) = (𝑃 ∖ {𝑧}))
16	15	unieqd 4446	. . . . . 6 ⊢ (𝑝 = 𝑃 → ∪ (𝑝 ∖ {𝑧}) = ∪ (𝑃 ∖ {𝑧}))
17	14, 16	ifbieq1d 4109	. . . . 5 ⊢ (𝑝 = 𝑃 → if(𝑧 ∈ 𝑝, ∪ (𝑝 ∖ {𝑧}), 𝑧) = if(𝑧 ∈ 𝑃, ∪ (𝑃 ∖ {𝑧}), 𝑧))
18	17	mpteq2dv 4745	. . . 4 ⊢ (𝑝 = 𝑃 → (𝑧 ∈ 𝐷 ↦ if(𝑧 ∈ 𝑝, ∪ (𝑝 ∖ {𝑧}), 𝑧)) = (𝑧 ∈ 𝐷 ↦ if(𝑧 ∈ 𝑃, ∪ (𝑃 ∖ {𝑧}), 𝑧)))
19		eqid 2622	. . . 4 ⊢ (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)))
20	18, 19	fvmptg 6280	. . 3 ⊢ ((𝑃 ∈ {𝑦 ∈ 𝒫 𝐷 ∣ 𝑦 ≈ 2𝑜} ∧ (𝑧 ∈ 𝐷 ↦ if(𝑧 ∈ 𝑃, ∪ (𝑃 ∖ {𝑧}), 𝑧)) ∈ V) → ((𝑝 ∈ {𝑦 ∈ 𝒫 𝐷 ∣ 𝑦 ≈ 2𝑜} ↦ (𝑧 ∈ 𝐷 ↦ if(𝑧 ∈ 𝑝, ∪ (𝑝 ∖ {𝑧}), 𝑧)))‘𝑃) = (𝑧 ∈ 𝐷 ↦ if(𝑧 ∈ 𝑃, ∪ (𝑃 ∖ {𝑧}), 𝑧)))
21	11, 13, 20	syl2anc 693	. 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)))
22	4, 21	eqtrd 2656	1 ⊢ ((𝐷 ∈ 𝑉 ∧ 𝑃 ⊆ 𝐷 ∧ 𝑃 ≈ 2𝑜) → (𝑇‘𝑃) = (𝑧 ∈ 𝐷 ↦ if(𝑧 ∈ 𝑃, ∪ (𝑃 ∖ {𝑧}), 𝑧)))

Syntax hints:   → wi 4   ∧ w3a 1037   = wceq 1483   ∈ wcel 1990  {crab 2916  Vcvv 3200   ∖ cdif 3571   ⊆ wss 3574  ifcif 4086  𝒫 cpw 4158  {csn 4177  ∪ cuni 4436   class class class wbr 4653   ↦ cmpt 4729  ‘cfv 5888  2_𝑜c2o 7554   ≈ cen 7952  pmTrspcpmtr 17861

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-pmtr 17862

This theorem is referenced by:  pmtrfv  17872  pmtrf  17875