Theorem pmtrfval 19583

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 3474	. . 3 ⊢ (𝐷 ∈ 𝑉 → 𝐷 ∈ V)
3		pweq 4574	. . . . . 6 ⊢ (𝑑 = 𝐷 → 𝒫 𝑑 = 𝒫 𝐷)
4	3	rabeqdv 3429	. . . . 5 ⊢ (𝑑 = 𝐷 → {𝑦 ∈ 𝒫 𝑑 ∣ 𝑦 ≈ 2o} = {𝑦 ∈ 𝒫 𝐷 ∣ 𝑦 ≈ 2o})
5		mpteq1 5198	. . . . 5 ⊢ (𝑑 = 𝐷 → (𝑧 ∈ 𝑑 ↦ if(𝑧 ∈ 𝑝, ∪ (𝑝 ∖ {𝑧}), 𝑧)) = (𝑧 ∈ 𝐷 ↦ if(𝑧 ∈ 𝑝, ∪ (𝑝 ∖ {𝑧}), 𝑧)))
6	4, 5	mpteq12dv 5196	. . . 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 19575	. . . 4 ⊢ pmTrsp = (𝑑 ∈ V ↦ (𝑝 ∈ {𝑦 ∈ 𝒫 𝑑 ∣ 𝑦 ≈ 2o} ↦ (𝑧 ∈ 𝑑 ↦ if(𝑧 ∈ 𝑝, ∪ (𝑝 ∖ {𝑧}), 𝑧))))
8		vpwex 5346	. . . . 5 ⊢ 𝒫 𝑑 ∈ V
9	8	mptrabex 7228	. . . 4 ⊢ (𝑝 ∈ {𝑦 ∈ 𝒫 𝑑 ∣ 𝑦 ≈ 2o} ↦ (𝑧 ∈ 𝑑 ↦ if(𝑧 ∈ 𝑝, ∪ (𝑝 ∖ {𝑧}), 𝑧))) ∈ V
10	6, 7, 9	fvmpt3i 6996	. . 3 ⊢ (𝐷 ∈ V → (pmTrsp‘𝐷) = (𝑝 ∈ {𝑦 ∈ 𝒫 𝐷 ∣ 𝑦 ≈ 2o} ↦ (𝑧 ∈ 𝐷 ↦ if(𝑧 ∈ 𝑝, ∪ (𝑝 ∖ {𝑧}), 𝑧))))
11	2, 10	syl 18	. 2 ⊢ (𝐷 ∈ 𝑉 → (pmTrsp‘𝐷) = (𝑝 ∈ {𝑦 ∈ 𝒫 𝐷 ∣ 𝑦 ≈ 2o} ↦ (𝑧 ∈ 𝐷 ↦ if(𝑧 ∈ 𝑝, ∪ (𝑝 ∖ {𝑧}), 𝑧))))
12	1, 11	eqtrid 2809	1 ⊢ (𝐷 ∈ 𝑉 → 𝑇 = (𝑝 ∈ {𝑦 ∈ 𝒫 𝐷 ∣ 𝑦 ≈ 2o} ↦ (𝑧 ∈ 𝐷 ↦ if(𝑧 ∈ 𝑝, ∪ (𝑝 ∖ {𝑧}), 𝑧))))

This proof depends on syntax axioms:   → wi 4   = wceq 1570   ∈ wcel 2145  {crab 3414  Vcvv 3453   ∖ cdif 3899  ifcif 4485  𝒫 cpw 4560  {csn 4587  ∪ cuni 4870   class class class wbr 5107   ↦ cmpt 5190  ‘cfv 6537  2_oc2o 8453   ≈ cen 8953  pmTrspcpmtr 19574

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2734  ax-rep 5236  ax-sep 5255  ax-nul 5267  ax-pow 5334  ax-pr 5402

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3079  df-rex 3089  df-reu 3368  df-rab 3415  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4283  df-if 4486  df-pw 4562  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4871  df-iun 4956  df-br 5108  df-opab 5172  df-mpt 5191  df-id 5554  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-pmtr 19575

This theorem is used by:  pmtrval  19584  pmtrrn  19590  pmtrfrn  19591  pmtrprfval  19620  pmtrsn  19652  trsp2cyc  33571