Theorem pmtrrn 19423

Description: Transposing two points gives a transposition function. (Contributed by Stefan O'Rear, 22-Aug-2015.)

Hypotheses

Ref	Expression
pmtrrn.t	⊢ 𝑇 = (pmTrsp‘𝐷)
pmtrrn.r	⊢ 𝑅 = ran 𝑇

Assertion

Ref	Expression
pmtrrn	⊢ ((𝐷 ∈ 𝑉 ∧ 𝑃 ⊆ 𝐷 ∧ 𝑃 ≈ 2o) → (𝑇‘𝑃) ∈ 𝑅)

Proof of Theorem pmtrrn

Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		mptexg 7169	. . . . . . 7 ⊢ (𝐷 ∈ 𝑉 → (𝑦 ∈ 𝐷 ↦ if(𝑦 ∈ 𝑧, ∪ (𝑧 ∖ {𝑦}), 𝑦)) ∈ V)
2	1	ralrimivw 3134	. . . . . 6 ⊢ (𝐷 ∈ 𝑉 → ∀𝑧 ∈ {𝑥 ∈ 𝒫 𝐷 ∣ 𝑥 ≈ 2o} (𝑦 ∈ 𝐷 ↦ if(𝑦 ∈ 𝑧, ∪ (𝑧 ∖ {𝑦}), 𝑦)) ∈ V)
3	2	3ad2ant1 1134	. . . . 5 ⊢ ((𝐷 ∈ 𝑉 ∧ 𝑃 ⊆ 𝐷 ∧ 𝑃 ≈ 2o) → ∀𝑧 ∈ {𝑥 ∈ 𝒫 𝐷 ∣ 𝑥 ≈ 2o} (𝑦 ∈ 𝐷 ↦ if(𝑦 ∈ 𝑧, ∪ (𝑧 ∖ {𝑦}), 𝑦)) ∈ V)
4		eqid 2737	. . . . . 6 ⊢ (𝑧 ∈ {𝑥 ∈ 𝒫 𝐷 ∣ 𝑥 ≈ 2o} ↦ (𝑦 ∈ 𝐷 ↦ if(𝑦 ∈ 𝑧, ∪ (𝑧 ∖ {𝑦}), 𝑦))) = (𝑧 ∈ {𝑥 ∈ 𝒫 𝐷 ∣ 𝑥 ≈ 2o} ↦ (𝑦 ∈ 𝐷 ↦ if(𝑦 ∈ 𝑧, ∪ (𝑧 ∖ {𝑦}), 𝑦)))
5	4	fnmpt 6632	. . . . 5 ⊢ (∀𝑧 ∈ {𝑥 ∈ 𝒫 𝐷 ∣ 𝑥 ≈ 2o} (𝑦 ∈ 𝐷 ↦ if(𝑦 ∈ 𝑧, ∪ (𝑧 ∖ {𝑦}), 𝑦)) ∈ V → (𝑧 ∈ {𝑥 ∈ 𝒫 𝐷 ∣ 𝑥 ≈ 2o} ↦ (𝑦 ∈ 𝐷 ↦ if(𝑦 ∈ 𝑧, ∪ (𝑧 ∖ {𝑦}), 𝑦))) Fn {𝑥 ∈ 𝒫 𝐷 ∣ 𝑥 ≈ 2o})
6	3, 5	syl 17	. . . 4 ⊢ ((𝐷 ∈ 𝑉 ∧ 𝑃 ⊆ 𝐷 ∧ 𝑃 ≈ 2o) → (𝑧 ∈ {𝑥 ∈ 𝒫 𝐷 ∣ 𝑥 ≈ 2o} ↦ (𝑦 ∈ 𝐷 ↦ if(𝑦 ∈ 𝑧, ∪ (𝑧 ∖ {𝑦}), 𝑦))) Fn {𝑥 ∈ 𝒫 𝐷 ∣ 𝑥 ≈ 2o})
7		pmtrrn.t	. . . . . . 7 ⊢ 𝑇 = (pmTrsp‘𝐷)
8	7	pmtrfval 19416	. . . . . 6 ⊢ (𝐷 ∈ 𝑉 → 𝑇 = (𝑧 ∈ {𝑥 ∈ 𝒫 𝐷 ∣ 𝑥 ≈ 2o} ↦ (𝑦 ∈ 𝐷 ↦ if(𝑦 ∈ 𝑧, ∪ (𝑧 ∖ {𝑦}), 𝑦))))
9	8	3ad2ant1 1134	. . . . 5 ⊢ ((𝐷 ∈ 𝑉 ∧ 𝑃 ⊆ 𝐷 ∧ 𝑃 ≈ 2o) → 𝑇 = (𝑧 ∈ {𝑥 ∈ 𝒫 𝐷 ∣ 𝑥 ≈ 2o} ↦ (𝑦 ∈ 𝐷 ↦ if(𝑦 ∈ 𝑧, ∪ (𝑧 ∖ {𝑦}), 𝑦))))
10	9	fneq1d 6585	. . . 4 ⊢ ((𝐷 ∈ 𝑉 ∧ 𝑃 ⊆ 𝐷 ∧ 𝑃 ≈ 2o) → (𝑇 Fn {𝑥 ∈ 𝒫 𝐷 ∣ 𝑥 ≈ 2o} ↔ (𝑧 ∈ {𝑥 ∈ 𝒫 𝐷 ∣ 𝑥 ≈ 2o} ↦ (𝑦 ∈ 𝐷 ↦ if(𝑦 ∈ 𝑧, ∪ (𝑧 ∖ {𝑦}), 𝑦))) Fn {𝑥 ∈ 𝒫 𝐷 ∣ 𝑥 ≈ 2o}))
11	6, 10	mpbird 257	. . 3 ⊢ ((𝐷 ∈ 𝑉 ∧ 𝑃 ⊆ 𝐷 ∧ 𝑃 ≈ 2o) → 𝑇 Fn {𝑥 ∈ 𝒫 𝐷 ∣ 𝑥 ≈ 2o})
12		breq1 5089	. . . 4 ⊢ (𝑥 = 𝑃 → (𝑥 ≈ 2o ↔ 𝑃 ≈ 2o))
13		elpw2g 5270	. . . . . 6 ⊢ (𝐷 ∈ 𝑉 → (𝑃 ∈ 𝒫 𝐷 ↔ 𝑃 ⊆ 𝐷))
14	13	biimpar 477	. . . . 5 ⊢ ((𝐷 ∈ 𝑉 ∧ 𝑃 ⊆ 𝐷) → 𝑃 ∈ 𝒫 𝐷)
15	14	3adant3 1133	. . . 4 ⊢ ((𝐷 ∈ 𝑉 ∧ 𝑃 ⊆ 𝐷 ∧ 𝑃 ≈ 2o) → 𝑃 ∈ 𝒫 𝐷)
16		simp3 1139	. . . 4 ⊢ ((𝐷 ∈ 𝑉 ∧ 𝑃 ⊆ 𝐷 ∧ 𝑃 ≈ 2o) → 𝑃 ≈ 2o)
17	12, 15, 16	elrabd 3637	. . 3 ⊢ ((𝐷 ∈ 𝑉 ∧ 𝑃 ⊆ 𝐷 ∧ 𝑃 ≈ 2o) → 𝑃 ∈ {𝑥 ∈ 𝒫 𝐷 ∣ 𝑥 ≈ 2o})
18		fnfvelrn 7026	. . 3 ⊢ ((𝑇 Fn {𝑥 ∈ 𝒫 𝐷 ∣ 𝑥 ≈ 2o} ∧ 𝑃 ∈ {𝑥 ∈ 𝒫 𝐷 ∣ 𝑥 ≈ 2o}) → (𝑇‘𝑃) ∈ ran 𝑇)
19	11, 17, 18	syl2anc 585	. 2 ⊢ ((𝐷 ∈ 𝑉 ∧ 𝑃 ⊆ 𝐷 ∧ 𝑃 ≈ 2o) → (𝑇‘𝑃) ∈ ran 𝑇)
20		pmtrrn.r	. 2 ⊢ 𝑅 = ran 𝑇
21	19, 20	eleqtrrdi 2848	1 ⊢ ((𝐷 ∈ 𝑉 ∧ 𝑃 ⊆ 𝐷 ∧ 𝑃 ≈ 2o) → (𝑇‘𝑃) ∈ 𝑅)

Syntax hints:   → wi 4   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114  ∀wral 3052  {crab 3390  Vcvv 3430   ∖ cdif 3887   ⊆ wss 3890  ifcif 4467  𝒫 cpw 4542  {csn 4568  ∪ cuni 4851   class class class wbr 5086   ↦ cmpt 5167  ran crn 5625   Fn wfn 6487  ‘cfv 6492  2_oc2o 8392   ≈ cen 8883  pmTrspcpmtr 19407

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 2709  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pow 5302  ax-pr 5370

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 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-iun 4936  df-br 5087  df-opab 5149  df-mpt 5168  df-id 5519  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-pmtr 19408

This theorem is referenced by:  pmtrfb  19431  symggen  19436  pmtr3ncom  19441  pmtrdifellem1  19442  mdetralt  22583  pmtrcnel  33165  pmtrcnel2  33166  fzo0pmtrlast  33168  pmtridf1o  33170  pmtrto1cl  33175  cyc3evpm  33226  cyc3genpmlem  33227  cyc3conja  33233