Theorem pmtrfval 19473

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 4568	. . . . . 6 ⊢ (𝑑 = 𝐷 → 𝒫 𝑑 = 𝒫 𝐷)
4	3	rabeqdv 3428	. . . . 5 ⊢ (𝑑 = 𝐷 → {𝑦 ∈ 𝒫 𝑑 ∣ 𝑦 ≈ 2o} = {𝑦 ∈ 𝒫 𝐷 ∣ 𝑦 ≈ 2o})
5		mpteq1 5188	. . . . 5 ⊢ (𝑑 = 𝐷 → (𝑧 ∈ 𝑑 ↦ if(𝑧 ∈ 𝑝, ∪ (𝑝 ∖ {𝑧}), 𝑧)) = (𝑧 ∈ 𝐷 ↦ if(𝑧 ∈ 𝑝, ∪ (𝑝 ∖ {𝑧}), 𝑧)))
6	4, 5	mpteq12dv 5186	. . . 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 19465	. . . 4 ⊢ pmTrsp = (𝑑 ∈ V ↦ (𝑝 ∈ {𝑦 ∈ 𝒫 𝑑 ∣ 𝑦 ≈ 2o} ↦ (𝑧 ∈ 𝑑 ↦ if(𝑧 ∈ 𝑝, ∪ (𝑝 ∖ {𝑧}), 𝑧))))
8		vpwex 5333	. . . . 5 ⊢ 𝒫 𝑑 ∈ V
9	8	mptrabex 7205	. . . 4 ⊢ (𝑝 ∈ {𝑦 ∈ 𝒫 𝑑 ∣ 𝑦 ≈ 2o} ↦ (𝑧 ∈ 𝑑 ↦ if(𝑧 ∈ 𝑝, ∪ (𝑝 ∖ {𝑧}), 𝑧))) ∈ V
10	6, 7, 9	fvmpt3i 6977	. . 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 2808	1 ⊢ (𝐷 ∈ 𝑉 → 𝑇 = (𝑝 ∈ {𝑦 ∈ 𝒫 𝐷 ∣ 𝑦 ≈ 2o} ↦ (𝑧 ∈ 𝐷 ↦ if(𝑧 ∈ 𝑝, ∪ (𝑝 ∖ {𝑧}), 𝑧))))

Syntax hints:   → wi 4   = wceq 1559   ∈ wcel 2141  {crab 3413  Vcvv 3453   ∖ cdif 3901  ifcif 4479  𝒫 cpw 4554  {csn 4581  ∪ cuni 4864   class class class wbr 5099   ↦ cmpt 5180  ‘cfv 6517  2_oc2o 8426   ≈ cen 8920  pmTrspcpmtr 19464

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-rep 5226  ax-sep 5245  ax-nul 5255  ax-pow 5321  ax-pr 5389

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3076  df-rex 3086  df-reu 3367  df-rab 3414  df-v 3455  df-sbc 3745  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-iun 4950  df-br 5100  df-opab 5162  df-mpt 5181  df-id 5540  df-xp 5651  df-rel 5652  df-cnv 5653  df-co 5654  df-dm 5655  df-rn 5656  df-res 5657  df-ima 5658  df-iota 6473  df-fun 6519  df-fn 6520  df-f 6521  df-f1 6522  df-fo 6523  df-f1o 6524  df-fv 6525  df-pmtr 19465

This theorem is referenced by:  pmtrval  19474  pmtrrn  19480  pmtrfrn  19481  pmtrprfval  19510  pmtrsn  19542  trsp2cyc  33264