Theorem pmtrrn 19370

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 7155	. . . . . . 7 ⊢ (𝐷 ∈ 𝑉 → (𝑦 ∈ 𝐷 ↦ if(𝑦 ∈ 𝑧, ∪ (𝑧 ∖ {𝑦}), 𝑦)) ∈ V)
2	1	ralrimivw 3128	. . . . . 6 ⊢ (𝐷 ∈ 𝑉 → ∀𝑧 ∈ {𝑥 ∈ 𝒫 𝐷 ∣ 𝑥 ≈ 2o} (𝑦 ∈ 𝐷 ↦ if(𝑦 ∈ 𝑧, ∪ (𝑧 ∖ {𝑦}), 𝑦)) ∈ V)
3	2	3ad2ant1 1133	. . . . 5 ⊢ ((𝐷 ∈ 𝑉 ∧ 𝑃 ⊆ 𝐷 ∧ 𝑃 ≈ 2o) → ∀𝑧 ∈ {𝑥 ∈ 𝒫 𝐷 ∣ 𝑥 ≈ 2o} (𝑦 ∈ 𝐷 ↦ if(𝑦 ∈ 𝑧, ∪ (𝑧 ∖ {𝑦}), 𝑦)) ∈ V)
4		eqid 2731	. . . . . 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 6621	. . . . 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 19363	. . . . . 6 ⊢ (𝐷 ∈ 𝑉 → 𝑇 = (𝑧 ∈ {𝑥 ∈ 𝒫 𝐷 ∣ 𝑥 ≈ 2o} ↦ (𝑦 ∈ 𝐷 ↦ if(𝑦 ∈ 𝑧, ∪ (𝑧 ∖ {𝑦}), 𝑦))))
9	8	3ad2ant1 1133	. . . . 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 6574	. . . 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 5094	. . . 4 ⊢ (𝑥 = 𝑃 → (𝑥 ≈ 2o ↔ 𝑃 ≈ 2o))
13		elpw2g 5271	. . . . . 6 ⊢ (𝐷 ∈ 𝑉 → (𝑃 ∈ 𝒫 𝐷 ↔ 𝑃 ⊆ 𝐷))
14	13	biimpar 477	. . . . 5 ⊢ ((𝐷 ∈ 𝑉 ∧ 𝑃 ⊆ 𝐷) → 𝑃 ∈ 𝒫 𝐷)
15	14	3adant3 1132	. . . 4 ⊢ ((𝐷 ∈ 𝑉 ∧ 𝑃 ⊆ 𝐷 ∧ 𝑃 ≈ 2o) → 𝑃 ∈ 𝒫 𝐷)
16		simp3 1138	. . . 4 ⊢ ((𝐷 ∈ 𝑉 ∧ 𝑃 ⊆ 𝐷 ∧ 𝑃 ≈ 2o) → 𝑃 ≈ 2o)
17	12, 15, 16	elrabd 3649	. . 3 ⊢ ((𝐷 ∈ 𝑉 ∧ 𝑃 ⊆ 𝐷 ∧ 𝑃 ≈ 2o) → 𝑃 ∈ {𝑥 ∈ 𝒫 𝐷 ∣ 𝑥 ≈ 2o})
18		fnfvelrn 7013	. . 3 ⊢ ((𝑇 Fn {𝑥 ∈ 𝒫 𝐷 ∣ 𝑥 ≈ 2o} ∧ 𝑃 ∈ {𝑥 ∈ 𝒫 𝐷 ∣ 𝑥 ≈ 2o}) → (𝑇‘𝑃) ∈ ran 𝑇)
19	11, 17, 18	syl2anc 584	. 2 ⊢ ((𝐷 ∈ 𝑉 ∧ 𝑃 ⊆ 𝐷 ∧ 𝑃 ≈ 2o) → (𝑇‘𝑃) ∈ ran 𝑇)
20		pmtrrn.r	. 2 ⊢ 𝑅 = ran 𝑇
21	19, 20	eleqtrrdi 2842	1 ⊢ ((𝐷 ∈ 𝑉 ∧ 𝑃 ⊆ 𝐷 ∧ 𝑃 ≈ 2o) → (𝑇‘𝑃) ∈ 𝑅)

Syntax hints:   → wi 4   ∧ w3a 1086   = wceq 1541   ∈ wcel 2111  ∀wral 3047  {crab 3395  Vcvv 3436   ∖ cdif 3899   ⊆ wss 3902  ifcif 4475  𝒫 cpw 4550  {csn 4576  ∪ cuni 4859   class class class wbr 5091   ↦ cmpt 5172  ran crn 5617   Fn wfn 6476  ‘cfv 6481  2_oc2o 8379   ≈ cen 8866  pmTrspcpmtr 19354

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-rep 5217  ax-sep 5234  ax-nul 5244  ax-pow 5303  ax-pr 5370

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-reu 3347  df-rab 3396  df-v 3438  df-sbc 3742  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4284  df-if 4476  df-pw 4552  df-sn 4577  df-pr 4579  df-op 4583  df-uni 4860  df-iun 4943  df-br 5092  df-opab 5154  df-mpt 5173  df-id 5511  df-xp 5622  df-rel 5623  df-cnv 5624  df-co 5625  df-dm 5626  df-rn 5627  df-res 5628  df-ima 5629  df-iota 6437  df-fun 6483  df-fn 6484  df-f 6485  df-f1 6486  df-fo 6487  df-f1o 6488  df-fv 6489  df-pmtr 19355

This theorem is referenced by:  pmtrfb  19378  symggen  19383  pmtr3ncom  19388  pmtrdifellem1  19389  mdetralt  22524  pmtrcnel  33056  pmtrcnel2  33057  fzo0pmtrlast  33059  pmtridf1o  33061  pmtrto1cl  33066  cyc3evpm  33117  cyc3genpmlem  33118  cyc3conja  33124