Theorem pmtrrn 17877

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 6484	. . . . . . 7 ⊢ (𝐷 ∈ 𝑉 → (𝑦 ∈ 𝐷 ↦ if(𝑦 ∈ 𝑧, ∪ (𝑧 ∖ {𝑦}), 𝑦)) ∈ V)
2	1	ralrimivw 2967	. . . . . 6 ⊢ (𝐷 ∈ 𝑉 → ∀𝑧 ∈ {𝑥 ∈ 𝒫 𝐷 ∣ 𝑥 ≈ 2𝑜} (𝑦 ∈ 𝐷 ↦ if(𝑦 ∈ 𝑧, ∪ (𝑧 ∖ {𝑦}), 𝑦)) ∈ V)
3	2	3ad2ant1 1082	. . . . 5 ⊢ ((𝐷 ∈ 𝑉 ∧ 𝑃 ⊆ 𝐷 ∧ 𝑃 ≈ 2𝑜) → ∀𝑧 ∈ {𝑥 ∈ 𝒫 𝐷 ∣ 𝑥 ≈ 2𝑜} (𝑦 ∈ 𝐷 ↦ if(𝑦 ∈ 𝑧, ∪ (𝑧 ∖ {𝑦}), 𝑦)) ∈ V)
4		eqid 2622	. . . . . 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 6020	. . . . 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 17870	. . . . . 6 ⊢ (𝐷 ∈ 𝑉 → 𝑇 = (𝑧 ∈ {𝑥 ∈ 𝒫 𝐷 ∣ 𝑥 ≈ 2𝑜} ↦ (𝑦 ∈ 𝐷 ↦ if(𝑦 ∈ 𝑧, ∪ (𝑧 ∖ {𝑦}), 𝑦))))
9	8	3ad2ant1 1082	. . . . 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 5981	. . . 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 247	. . 3 ⊢ ((𝐷 ∈ 𝑉 ∧ 𝑃 ⊆ 𝐷 ∧ 𝑃 ≈ 2𝑜) → 𝑇 Fn {𝑥 ∈ 𝒫 𝐷 ∣ 𝑥 ≈ 2𝑜})
12		elpw2g 4827	. . . . . 6 ⊢ (𝐷 ∈ 𝑉 → (𝑃 ∈ 𝒫 𝐷 ↔ 𝑃 ⊆ 𝐷))
13	12	biimpar 502	. . . . 5 ⊢ ((𝐷 ∈ 𝑉 ∧ 𝑃 ⊆ 𝐷) → 𝑃 ∈ 𝒫 𝐷)
14	13	3adant3 1081	. . . 4 ⊢ ((𝐷 ∈ 𝑉 ∧ 𝑃 ⊆ 𝐷 ∧ 𝑃 ≈ 2𝑜) → 𝑃 ∈ 𝒫 𝐷)
15		simp3 1063	. . . 4 ⊢ ((𝐷 ∈ 𝑉 ∧ 𝑃 ⊆ 𝐷 ∧ 𝑃 ≈ 2𝑜) → 𝑃 ≈ 2𝑜)
16		breq1 4656	. . . . 5 ⊢ (𝑥 = 𝑃 → (𝑥 ≈ 2𝑜 ↔ 𝑃 ≈ 2𝑜))
17	16	elrab 3363	. . . 4 ⊢ (𝑃 ∈ {𝑥 ∈ 𝒫 𝐷 ∣ 𝑥 ≈ 2𝑜} ↔ (𝑃 ∈ 𝒫 𝐷 ∧ 𝑃 ≈ 2𝑜))
18	14, 15, 17	sylanbrc 698	. . 3 ⊢ ((𝐷 ∈ 𝑉 ∧ 𝑃 ⊆ 𝐷 ∧ 𝑃 ≈ 2𝑜) → 𝑃 ∈ {𝑥 ∈ 𝒫 𝐷 ∣ 𝑥 ≈ 2𝑜})
19		fnfvelrn 6356	. . 3 ⊢ ((𝑇 Fn {𝑥 ∈ 𝒫 𝐷 ∣ 𝑥 ≈ 2𝑜} ∧ 𝑃 ∈ {𝑥 ∈ 𝒫 𝐷 ∣ 𝑥 ≈ 2𝑜}) → (𝑇‘𝑃) ∈ ran 𝑇)
20	11, 18, 19	syl2anc 693	. 2 ⊢ ((𝐷 ∈ 𝑉 ∧ 𝑃 ⊆ 𝐷 ∧ 𝑃 ≈ 2𝑜) → (𝑇‘𝑃) ∈ ran 𝑇)
21		pmtrrn.r	. 2 ⊢ 𝑅 = ran 𝑇
22	20, 21	syl6eleqr 2712	1 ⊢ ((𝐷 ∈ 𝑉 ∧ 𝑃 ⊆ 𝐷 ∧ 𝑃 ≈ 2𝑜) → (𝑇‘𝑃) ∈ 𝑅)

Syntax hints:   → wi 4   ∧ w3a 1037   = wceq 1483   ∈ wcel 1990  ∀wral 2912  {crab 2916  Vcvv 3200   ∖ cdif 3571   ⊆ wss 3574  ifcif 4086  𝒫 cpw 4158  {csn 4177  ∪ cuni 4436   class class class wbr 4653   ↦ cmpt 4729  ran crn 5115   Fn wfn 5883  ‘cfv 5888  2_𝑜c2o 7554   ≈ cen 7952  pmTrspcpmtr 17861

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-pmtr 17862

This theorem is referenced by:  pmtrfb  17885  symggen  17890  pmtr3ncom  17895  pmtrdifellem1  17896  mdetralt  20414  pmtrto1cl  29849  pmtridf1o  29856