Theorem pmtrfval 19425

Description: The function generating transpositions on a set. (Contributed by Stefan O'Rear, 16-Aug-2015.)

Hypothesis

Ref	Expression
pmtrfval.t	⊢ 𝑇 = (pmTrsp‘𝐷)

Assertion

Ref	Expression
pmtrfval	⊢ (𝐷 ∈ 𝑉 → 𝑇 = (𝑝 ∈ {𝑦 ∈ 𝒫 𝐷 ∣ 𝑦 ≈ 2o} ↦ (𝑧 ∈ 𝐷 ↦ if(𝑧 ∈ 𝑝, ∪ (𝑝 ∖ {𝑧}), 𝑧))))

Distinct variable groups:   𝑦,𝑝,𝑧,𝐷   𝑇,𝑝,𝑦,𝑧   𝑧,𝑉

Allowed substitution hints:   𝑉(𝑦,𝑝)

Proof of Theorem pmtrfval

Dummy variable 𝑑 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		pmtrfval.t	. 2 ⊢ 𝑇 = (pmTrsp‘𝐷)
2		elex 3450	. . 3 ⊢ (𝐷 ∈ 𝑉 → 𝐷 ∈ V)
3		pweq 4555	. . . . . 6 ⊢ (𝑑 = 𝐷 → 𝒫 𝑑 = 𝒫 𝐷)
4	3	rabeqdv 3404	. . . . 5 ⊢ (𝑑 = 𝐷 → {𝑦 ∈ 𝒫 𝑑 ∣ 𝑦 ≈ 2o} = {𝑦 ∈ 𝒫 𝐷 ∣ 𝑦 ≈ 2o})
5		mpteq1 5174	. . . . 5 ⊢ (𝑑 = 𝐷 → (𝑧 ∈ 𝑑 ↦ if(𝑧 ∈ 𝑝, ∪ (𝑝 ∖ {𝑧}), 𝑧)) = (𝑧 ∈ 𝐷 ↦ if(𝑧 ∈ 𝑝, ∪ (𝑝 ∖ {𝑧}), 𝑧)))
6	4, 5	mpteq12dv 5172	. . . 4 ⊢ (𝑑 = 𝐷 → (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))))
7		df-pmtr 19417	. . . 4 ⊢ pmTrsp = (𝑑 ∈ V ↦ (𝑝 ∈ {𝑦 ∈ 𝒫 𝑑 ∣ 𝑦 ≈ 2o} ↦ (𝑧 ∈ 𝑑 ↦ if(𝑧 ∈ 𝑝, ∪ (𝑝 ∖ {𝑧}), 𝑧))))
8		vpwex 5319	. . . . 5 ⊢ 𝒫 𝑑 ∈ V
9	8	mptrabex 7180	. . . 4 ⊢ (𝑝 ∈ {𝑦 ∈ 𝒫 𝑑 ∣ 𝑦 ≈ 2o} ↦ (𝑧 ∈ 𝑑 ↦ if(𝑧 ∈ 𝑝, ∪ (𝑝 ∖ {𝑧}), 𝑧))) ∈ V
10	6, 7, 9	fvmpt3i 6953	. . 3 ⊢ (𝐷 ∈ V → (pmTrsp‘𝐷) = (𝑝 ∈ {𝑦 ∈ 𝒫 𝐷 ∣ 𝑦 ≈ 2o} ↦ (𝑧 ∈ 𝐷 ↦ if(𝑧 ∈ 𝑝, ∪ (𝑝 ∖ {𝑧}), 𝑧))))
11	2, 10	syl 17	. 2 ⊢ (𝐷 ∈ 𝑉 → (pmTrsp‘𝐷) = (𝑝 ∈ {𝑦 ∈ 𝒫 𝐷 ∣ 𝑦 ≈ 2o} ↦ (𝑧 ∈ 𝐷 ↦ if(𝑧 ∈ 𝑝, ∪ (𝑝 ∖ {𝑧}), 𝑧))))
12	1, 11	eqtrid 2783	1 ⊢ (𝐷 ∈ 𝑉 → 𝑇 = (𝑝 ∈ {𝑦 ∈ 𝒫 𝐷 ∣ 𝑦 ≈ 2o} ↦ (𝑧 ∈ 𝐷 ↦ if(𝑧 ∈ 𝑝, ∪ (𝑝 ∖ {𝑧}), 𝑧))))

Syntax hints:   → wi 4   = wceq 1542   ∈ wcel 2114  {crab 3389  Vcvv 3429   ∖ cdif 3886  ifcif 4466  𝒫 cpw 4541  {csn 4567  ∪ cuni 4850   class class class wbr 5085   ↦ cmpt 5166  ‘cfv 6498  2_oc2o 8399   ≈ cen 8890  pmTrspcpmtr 19416

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pow 5307  ax-pr 5375

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3062  df-reu 3343  df-rab 3390  df-v 3431  df-sbc 3729  df-csb 3838  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-iun 4935  df-br 5086  df-opab 5148  df-mpt 5167  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-iota 6454  df-fun 6500  df-fn 6501  df-f 6502  df-f1 6503  df-fo 6504  df-f1o 6505  df-fv 6506  df-pmtr 19417

This theorem is referenced by:  pmtrval  19426  pmtrrn  19432  pmtrfrn  19433  pmtrprfval  19462  pmtrsn  19494  trsp2cyc  33184