Theorem pmtrrn 19432

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 7176	. . . . . . 7 ⊢ (𝐷 ∈ 𝑉 → (𝑦 ∈ 𝐷 ↦ if(𝑦 ∈ 𝑧, ∪ (𝑧 ∖ {𝑦}), 𝑦)) ∈ V)
2	1	ralrimivw 3133	. . . . . 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 2736	. . . . . 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 6638	. . . . 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 19425	. . . . . 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 6591	. . . 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 5088	. . . 4 ⊢ (𝑥 = 𝑃 → (𝑥 ≈ 2o ↔ 𝑃 ≈ 2o))
13		elpw2g 5274	. . . . . 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 3636	. . 3 ⊢ ((𝐷 ∈ 𝑉 ∧ 𝑃 ⊆ 𝐷 ∧ 𝑃 ≈ 2o) → 𝑃 ∈ {𝑥 ∈ 𝒫 𝐷 ∣ 𝑥 ≈ 2o})
18		fnfvelrn 7032	. . 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 2847	1 ⊢ ((𝐷 ∈ 𝑉 ∧ 𝑃 ⊆ 𝐷 ∧ 𝑃 ≈ 2o) → (𝑇‘𝑃) ∈ 𝑅)

Syntax hints:   → wi 4   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114  ∀wral 3051  {crab 3389  Vcvv 3429   ∖ cdif 3886   ⊆ wss 3889  ifcif 4466  𝒫 cpw 4541  {csn 4567  ∪ cuni 4850   class class class wbr 5085   ↦ cmpt 5166  ran crn 5632   Fn wfn 6493  ‘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:  pmtrfb  19440  symggen  19445  pmtr3ncom  19450  pmtrdifellem1  19451  mdetralt  22573  pmtrcnel  33150  pmtrcnel2  33151  fzo0pmtrlast  33153  pmtridf1o  33155  pmtrto1cl  33160  cyc3evpm  33211  cyc3genpmlem  33212  cyc3conja  33218