Theorem pmtrrn 18980

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 7079	. . . . . . 7 ⊢ (𝐷 ∈ 𝑉 → (𝑦 ∈ 𝐷 ↦ if(𝑦 ∈ 𝑧, ∪ (𝑧 ∖ {𝑦}), 𝑦)) ∈ V)
2	1	ralrimivw 3108	. . . . . 6 ⊢ (𝐷 ∈ 𝑉 → ∀𝑧 ∈ {𝑥 ∈ 𝒫 𝐷 ∣ 𝑥 ≈ 2o} (𝑦 ∈ 𝐷 ↦ if(𝑦 ∈ 𝑧, ∪ (𝑧 ∖ {𝑦}), 𝑦)) ∈ V)
3	2	3ad2ant1 1131	. . . . 5 ⊢ ((𝐷 ∈ 𝑉 ∧ 𝑃 ⊆ 𝐷 ∧ 𝑃 ≈ 2o) → ∀𝑧 ∈ {𝑥 ∈ 𝒫 𝐷 ∣ 𝑥 ≈ 2o} (𝑦 ∈ 𝐷 ↦ if(𝑦 ∈ 𝑧, ∪ (𝑧 ∖ {𝑦}), 𝑦)) ∈ V)
4		eqid 2738	. . . . . 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 6557	. . . . 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 18973	. . . . . 6 ⊢ (𝐷 ∈ 𝑉 → 𝑇 = (𝑧 ∈ {𝑥 ∈ 𝒫 𝐷 ∣ 𝑥 ≈ 2o} ↦ (𝑦 ∈ 𝐷 ↦ if(𝑦 ∈ 𝑧, ∪ (𝑧 ∖ {𝑦}), 𝑦))))
9	8	3ad2ant1 1131	. . . . 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 6510	. . . 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 256	. . 3 ⊢ ((𝐷 ∈ 𝑉 ∧ 𝑃 ⊆ 𝐷 ∧ 𝑃 ≈ 2o) → 𝑇 Fn {𝑥 ∈ 𝒫 𝐷 ∣ 𝑥 ≈ 2o})
12		breq1 5073	. . . 4 ⊢ (𝑥 = 𝑃 → (𝑥 ≈ 2o ↔ 𝑃 ≈ 2o))
13		elpw2g 5263	. . . . . 6 ⊢ (𝐷 ∈ 𝑉 → (𝑃 ∈ 𝒫 𝐷 ↔ 𝑃 ⊆ 𝐷))
14	13	biimpar 477	. . . . 5 ⊢ ((𝐷 ∈ 𝑉 ∧ 𝑃 ⊆ 𝐷) → 𝑃 ∈ 𝒫 𝐷)
15	14	3adant3 1130	. . . 4 ⊢ ((𝐷 ∈ 𝑉 ∧ 𝑃 ⊆ 𝐷 ∧ 𝑃 ≈ 2o) → 𝑃 ∈ 𝒫 𝐷)
16		simp3 1136	. . . 4 ⊢ ((𝐷 ∈ 𝑉 ∧ 𝑃 ⊆ 𝐷 ∧ 𝑃 ≈ 2o) → 𝑃 ≈ 2o)
17	12, 15, 16	elrabd 3619	. . 3 ⊢ ((𝐷 ∈ 𝑉 ∧ 𝑃 ⊆ 𝐷 ∧ 𝑃 ≈ 2o) → 𝑃 ∈ {𝑥 ∈ 𝒫 𝐷 ∣ 𝑥 ≈ 2o})
18		fnfvelrn 6940	. . 3 ⊢ ((𝑇 Fn {𝑥 ∈ 𝒫 𝐷 ∣ 𝑥 ≈ 2o} ∧ 𝑃 ∈ {𝑥 ∈ 𝒫 𝐷 ∣ 𝑥 ≈ 2o}) → (𝑇‘𝑃) ∈ ran 𝑇)
19	11, 17, 18	syl2anc 583	. 2 ⊢ ((𝐷 ∈ 𝑉 ∧ 𝑃 ⊆ 𝐷 ∧ 𝑃 ≈ 2o) → (𝑇‘𝑃) ∈ ran 𝑇)
20		pmtrrn.r	. 2 ⊢ 𝑅 = ran 𝑇
21	19, 20	eleqtrrdi 2850	1 ⊢ ((𝐷 ∈ 𝑉 ∧ 𝑃 ⊆ 𝐷 ∧ 𝑃 ≈ 2o) → (𝑇‘𝑃) ∈ 𝑅)

Syntax hints:   → wi 4   ∧ w3a 1085   = wceq 1539   ∈ wcel 2108  ∀wral 3063  {crab 3067  Vcvv 3422   ∖ cdif 3880   ⊆ wss 3883  ifcif 4456  𝒫 cpw 4530  {csn 4558  ∪ cuni 4836   class class class wbr 5070   ↦ cmpt 5153  ran crn 5581   Fn wfn 6413  ‘cfv 6418  2_oc2o 8261   ≈ cen 8688  pmTrspcpmtr 18964

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-rep 5205  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-reu 3070  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-iun 4923  df-br 5071  df-opab 5133  df-mpt 5154  df-id 5480  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-pmtr 18965

This theorem is referenced by:  pmtrfb  18988  symggen  18993  pmtr3ncom  18998  pmtrdifellem1  18999  mdetralt  21665  pmtrcnel  31260  pmtrcnel2  31261  pmtridf1o  31263  pmtrto1cl  31268  cyc3evpm  31319  cyc3genpmlem  31320  cyc3conja  31326