Theorem pmtrfrn 17878

Description: A transposition (as a kind of function) is the function transposing the two points it moves. (Contributed by Stefan O'Rear, 22-Aug-2015.)

Hypotheses

Ref	Expression
pmtrrn.t	⊢ 𝑇 = (pmTrsp‘𝐷)
pmtrrn.r	⊢ 𝑅 = ran 𝑇
pmtrfrn.p	⊢ 𝑃 = dom (𝐹 ∖ I )

Assertion

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

Proof of Theorem pmtrfrn

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

Step	Hyp	Ref	Expression
1		noel 3919	. . . 4 ⊢ ¬ 𝐹 ∈ ∅
2		pmtrrn.r	. . . . . 6 ⊢ 𝑅 = ran 𝑇
3		pmtrrn.t	. . . . . . . . 9 ⊢ 𝑇 = (pmTrsp‘𝐷)
4		fvprc 6185	. . . . . . . . 9 ⊢ (¬ 𝐷 ∈ V → (pmTrsp‘𝐷) = ∅)
5	3, 4	syl5eq 2668	. . . . . . . 8 ⊢ (¬ 𝐷 ∈ V → 𝑇 = ∅)
6	5	rneqd 5353	. . . . . . 7 ⊢ (¬ 𝐷 ∈ V → ran 𝑇 = ran ∅)
7		rn0 5377	. . . . . . 7 ⊢ ran ∅ = ∅
8	6, 7	syl6eq 2672	. . . . . 6 ⊢ (¬ 𝐷 ∈ V → ran 𝑇 = ∅)
9	2, 8	syl5eq 2668	. . . . 5 ⊢ (¬ 𝐷 ∈ V → 𝑅 = ∅)
10	9	eleq2d 2687	. . . 4 ⊢ (¬ 𝐷 ∈ V → (𝐹 ∈ 𝑅 ↔ 𝐹 ∈ ∅))
11	1, 10	mtbiri 317	. . 3 ⊢ (¬ 𝐷 ∈ V → ¬ 𝐹 ∈ 𝑅)
12	11	con4i 113	. 2 ⊢ (𝐹 ∈ 𝑅 → 𝐷 ∈ V)
13		mptexg 6484	. . . . . . . 8 ⊢ (𝐷 ∈ V → (𝑧 ∈ 𝐷 ↦ if(𝑧 ∈ 𝑤, ∪ (𝑤 ∖ {𝑧}), 𝑧)) ∈ V)
14	13	ralrimivw 2967	. . . . . . 7 ⊢ (𝐷 ∈ V → ∀𝑤 ∈ {𝑥 ∈ 𝒫 𝐷 ∣ 𝑥 ≈ 2𝑜} (𝑧 ∈ 𝐷 ↦ if(𝑧 ∈ 𝑤, ∪ (𝑤 ∖ {𝑧}), 𝑧)) ∈ V)
15		eqid 2622	. . . . . . . 8 ⊢ (𝑤 ∈ {𝑥 ∈ 𝒫 𝐷 ∣ 𝑥 ≈ 2𝑜} ↦ (𝑧 ∈ 𝐷 ↦ if(𝑧 ∈ 𝑤, ∪ (𝑤 ∖ {𝑧}), 𝑧))) = (𝑤 ∈ {𝑥 ∈ 𝒫 𝐷 ∣ 𝑥 ≈ 2𝑜} ↦ (𝑧 ∈ 𝐷 ↦ if(𝑧 ∈ 𝑤, ∪ (𝑤 ∖ {𝑧}), 𝑧)))
16	15	fnmpt 6020	. . . . . . 7 ⊢ (∀𝑤 ∈ {𝑥 ∈ 𝒫 𝐷 ∣ 𝑥 ≈ 2𝑜} (𝑧 ∈ 𝐷 ↦ if(𝑧 ∈ 𝑤, ∪ (𝑤 ∖ {𝑧}), 𝑧)) ∈ V → (𝑤 ∈ {𝑥 ∈ 𝒫 𝐷 ∣ 𝑥 ≈ 2𝑜} ↦ (𝑧 ∈ 𝐷 ↦ if(𝑧 ∈ 𝑤, ∪ (𝑤 ∖ {𝑧}), 𝑧))) Fn {𝑥 ∈ 𝒫 𝐷 ∣ 𝑥 ≈ 2𝑜})
17	14, 16	syl 17	. . . . . 6 ⊢ (𝐷 ∈ V → (𝑤 ∈ {𝑥 ∈ 𝒫 𝐷 ∣ 𝑥 ≈ 2𝑜} ↦ (𝑧 ∈ 𝐷 ↦ if(𝑧 ∈ 𝑤, ∪ (𝑤 ∖ {𝑧}), 𝑧))) Fn {𝑥 ∈ 𝒫 𝐷 ∣ 𝑥 ≈ 2𝑜})
18	3	pmtrfval 17870	. . . . . . 7 ⊢ (𝐷 ∈ V → 𝑇 = (𝑤 ∈ {𝑥 ∈ 𝒫 𝐷 ∣ 𝑥 ≈ 2𝑜} ↦ (𝑧 ∈ 𝐷 ↦ if(𝑧 ∈ 𝑤, ∪ (𝑤 ∖ {𝑧}), 𝑧))))
19	18	fneq1d 5981	. . . . . 6 ⊢ (𝐷 ∈ V → (𝑇 Fn {𝑥 ∈ 𝒫 𝐷 ∣ 𝑥 ≈ 2𝑜} ↔ (𝑤 ∈ {𝑥 ∈ 𝒫 𝐷 ∣ 𝑥 ≈ 2𝑜} ↦ (𝑧 ∈ 𝐷 ↦ if(𝑧 ∈ 𝑤, ∪ (𝑤 ∖ {𝑧}), 𝑧))) Fn {𝑥 ∈ 𝒫 𝐷 ∣ 𝑥 ≈ 2𝑜}))
20	17, 19	mpbird 247	. . . . 5 ⊢ (𝐷 ∈ V → 𝑇 Fn {𝑥 ∈ 𝒫 𝐷 ∣ 𝑥 ≈ 2𝑜})
21		fvelrnb 6243	. . . . 5 ⊢ (𝑇 Fn {𝑥 ∈ 𝒫 𝐷 ∣ 𝑥 ≈ 2𝑜} → (𝐹 ∈ ran 𝑇 ↔ ∃𝑦 ∈ {𝑥 ∈ 𝒫 𝐷 ∣ 𝑥 ≈ 2𝑜} (𝑇‘𝑦) = 𝐹))
22	20, 21	syl 17	. . . 4 ⊢ (𝐷 ∈ V → (𝐹 ∈ ran 𝑇 ↔ ∃𝑦 ∈ {𝑥 ∈ 𝒫 𝐷 ∣ 𝑥 ≈ 2𝑜} (𝑇‘𝑦) = 𝐹))
23	2	eleq2i 2693	. . . 4 ⊢ (𝐹 ∈ 𝑅 ↔ 𝐹 ∈ ran 𝑇)
24		breq1 4656	. . . . . 6 ⊢ (𝑥 = 𝑦 → (𝑥 ≈ 2𝑜 ↔ 𝑦 ≈ 2𝑜))
25	24	rexrab 3370	. . . . 5 ⊢ (∃𝑦 ∈ {𝑥 ∈ 𝒫 𝐷 ∣ 𝑥 ≈ 2𝑜} (𝑇‘𝑦) = 𝐹 ↔ ∃𝑦 ∈ 𝒫 𝐷(𝑦 ≈ 2𝑜 ∧ (𝑇‘𝑦) = 𝐹))
26	25	bicomi 214	. . . 4 ⊢ (∃𝑦 ∈ 𝒫 𝐷(𝑦 ≈ 2𝑜 ∧ (𝑇‘𝑦) = 𝐹) ↔ ∃𝑦 ∈ {𝑥 ∈ 𝒫 𝐷 ∣ 𝑥 ≈ 2𝑜} (𝑇‘𝑦) = 𝐹)
27	22, 23, 26	3bitr4g 303	. . 3 ⊢ (𝐷 ∈ V → (𝐹 ∈ 𝑅 ↔ ∃𝑦 ∈ 𝒫 𝐷(𝑦 ≈ 2𝑜 ∧ (𝑇‘𝑦) = 𝐹)))
28		elpwi 4168	. . . . 5 ⊢ (𝑦 ∈ 𝒫 𝐷 → 𝑦 ⊆ 𝐷)
29		simp1 1061	. . . . . . . . . 10 ⊢ ((𝐷 ∈ V ∧ 𝑦 ⊆ 𝐷 ∧ 𝑦 ≈ 2𝑜) → 𝐷 ∈ V)
30	3	pmtrmvd 17876	. . . . . . . . . . 11 ⊢ ((𝐷 ∈ V ∧ 𝑦 ⊆ 𝐷 ∧ 𝑦 ≈ 2𝑜) → dom ((𝑇‘𝑦) ∖ I ) = 𝑦)
31		simp2 1062	. . . . . . . . . . 11 ⊢ ((𝐷 ∈ V ∧ 𝑦 ⊆ 𝐷 ∧ 𝑦 ≈ 2𝑜) → 𝑦 ⊆ 𝐷)
32	30, 31	eqsstrd 3639	. . . . . . . . . 10 ⊢ ((𝐷 ∈ V ∧ 𝑦 ⊆ 𝐷 ∧ 𝑦 ≈ 2𝑜) → dom ((𝑇‘𝑦) ∖ I ) ⊆ 𝐷)
33		simp3 1063	. . . . . . . . . . 11 ⊢ ((𝐷 ∈ V ∧ 𝑦 ⊆ 𝐷 ∧ 𝑦 ≈ 2𝑜) → 𝑦 ≈ 2𝑜)
34	30, 33	eqbrtrd 4675	. . . . . . . . . 10 ⊢ ((𝐷 ∈ V ∧ 𝑦 ⊆ 𝐷 ∧ 𝑦 ≈ 2𝑜) → dom ((𝑇‘𝑦) ∖ I ) ≈ 2𝑜)
35	29, 32, 34	3jca 1242	. . . . . . . . 9 ⊢ ((𝐷 ∈ V ∧ 𝑦 ⊆ 𝐷 ∧ 𝑦 ≈ 2𝑜) → (𝐷 ∈ V ∧ dom ((𝑇‘𝑦) ∖ I ) ⊆ 𝐷 ∧ dom ((𝑇‘𝑦) ∖ I ) ≈ 2𝑜))
36	30	eqcomd 2628	. . . . . . . . . 10 ⊢ ((𝐷 ∈ V ∧ 𝑦 ⊆ 𝐷 ∧ 𝑦 ≈ 2𝑜) → 𝑦 = dom ((𝑇‘𝑦) ∖ I ))
37	36	fveq2d 6195	. . . . . . . . 9 ⊢ ((𝐷 ∈ V ∧ 𝑦 ⊆ 𝐷 ∧ 𝑦 ≈ 2𝑜) → (𝑇‘𝑦) = (𝑇‘dom ((𝑇‘𝑦) ∖ I )))
38	35, 37	jca 554	. . . . . . . 8 ⊢ ((𝐷 ∈ V ∧ 𝑦 ⊆ 𝐷 ∧ 𝑦 ≈ 2𝑜) → ((𝐷 ∈ V ∧ dom ((𝑇‘𝑦) ∖ I ) ⊆ 𝐷 ∧ dom ((𝑇‘𝑦) ∖ I ) ≈ 2𝑜) ∧ (𝑇‘𝑦) = (𝑇‘dom ((𝑇‘𝑦) ∖ I ))))
39		difeq1 3721	. . . . . . . . . . 11 ⊢ ((𝑇‘𝑦) = 𝐹 → ((𝑇‘𝑦) ∖ I ) = (𝐹 ∖ I ))
40	39	dmeqd 5326	. . . . . . . . . 10 ⊢ ((𝑇‘𝑦) = 𝐹 → dom ((𝑇‘𝑦) ∖ I ) = dom (𝐹 ∖ I ))
41		pmtrfrn.p	. . . . . . . . . 10 ⊢ 𝑃 = dom (𝐹 ∖ I )
42	40, 41	syl6eqr 2674	. . . . . . . . 9 ⊢ ((𝑇‘𝑦) = 𝐹 → dom ((𝑇‘𝑦) ∖ I ) = 𝑃)
43		sseq1 3626	. . . . . . . . . . . 12 ⊢ (dom ((𝑇‘𝑦) ∖ I ) = 𝑃 → (dom ((𝑇‘𝑦) ∖ I ) ⊆ 𝐷 ↔ 𝑃 ⊆ 𝐷))
44		breq1 4656	. . . . . . . . . . . 12 ⊢ (dom ((𝑇‘𝑦) ∖ I ) = 𝑃 → (dom ((𝑇‘𝑦) ∖ I ) ≈ 2𝑜 ↔ 𝑃 ≈ 2𝑜))
45	43, 44	3anbi23d 1402	. . . . . . . . . . 11 ⊢ (dom ((𝑇‘𝑦) ∖ I ) = 𝑃 → ((𝐷 ∈ V ∧ dom ((𝑇‘𝑦) ∖ I ) ⊆ 𝐷 ∧ dom ((𝑇‘𝑦) ∖ I ) ≈ 2𝑜) ↔ (𝐷 ∈ V ∧ 𝑃 ⊆ 𝐷 ∧ 𝑃 ≈ 2𝑜)))
46	45	adantl 482	. . . . . . . . . 10 ⊢ (((𝑇‘𝑦) = 𝐹 ∧ dom ((𝑇‘𝑦) ∖ I ) = 𝑃) → ((𝐷 ∈ V ∧ dom ((𝑇‘𝑦) ∖ I ) ⊆ 𝐷 ∧ dom ((𝑇‘𝑦) ∖ I ) ≈ 2𝑜) ↔ (𝐷 ∈ V ∧ 𝑃 ⊆ 𝐷 ∧ 𝑃 ≈ 2𝑜)))
47		simpl 473	. . . . . . . . . . 11 ⊢ (((𝑇‘𝑦) = 𝐹 ∧ dom ((𝑇‘𝑦) ∖ I ) = 𝑃) → (𝑇‘𝑦) = 𝐹)
48		fveq2 6191	. . . . . . . . . . . 12 ⊢ (dom ((𝑇‘𝑦) ∖ I ) = 𝑃 → (𝑇‘dom ((𝑇‘𝑦) ∖ I )) = (𝑇‘𝑃))
49	48	adantl 482	. . . . . . . . . . 11 ⊢ (((𝑇‘𝑦) = 𝐹 ∧ dom ((𝑇‘𝑦) ∖ I ) = 𝑃) → (𝑇‘dom ((𝑇‘𝑦) ∖ I )) = (𝑇‘𝑃))
50	47, 49	eqeq12d 2637	. . . . . . . . . 10 ⊢ (((𝑇‘𝑦) = 𝐹 ∧ dom ((𝑇‘𝑦) ∖ I ) = 𝑃) → ((𝑇‘𝑦) = (𝑇‘dom ((𝑇‘𝑦) ∖ I )) ↔ 𝐹 = (𝑇‘𝑃)))
51	46, 50	anbi12d 747	. . . . . . . . 9 ⊢ (((𝑇‘𝑦) = 𝐹 ∧ dom ((𝑇‘𝑦) ∖ I ) = 𝑃) → (((𝐷 ∈ V ∧ dom ((𝑇‘𝑦) ∖ I ) ⊆ 𝐷 ∧ dom ((𝑇‘𝑦) ∖ I ) ≈ 2𝑜) ∧ (𝑇‘𝑦) = (𝑇‘dom ((𝑇‘𝑦) ∖ I ))) ↔ ((𝐷 ∈ V ∧ 𝑃 ⊆ 𝐷 ∧ 𝑃 ≈ 2𝑜) ∧ 𝐹 = (𝑇‘𝑃))))
52	42, 51	mpdan 702	. . . . . . . 8 ⊢ ((𝑇‘𝑦) = 𝐹 → (((𝐷 ∈ V ∧ dom ((𝑇‘𝑦) ∖ I ) ⊆ 𝐷 ∧ dom ((𝑇‘𝑦) ∖ I ) ≈ 2𝑜) ∧ (𝑇‘𝑦) = (𝑇‘dom ((𝑇‘𝑦) ∖ I ))) ↔ ((𝐷 ∈ V ∧ 𝑃 ⊆ 𝐷 ∧ 𝑃 ≈ 2𝑜) ∧ 𝐹 = (𝑇‘𝑃))))
53	38, 52	syl5ibcom 235	. . . . . . 7 ⊢ ((𝐷 ∈ V ∧ 𝑦 ⊆ 𝐷 ∧ 𝑦 ≈ 2𝑜) → ((𝑇‘𝑦) = 𝐹 → ((𝐷 ∈ V ∧ 𝑃 ⊆ 𝐷 ∧ 𝑃 ≈ 2𝑜) ∧ 𝐹 = (𝑇‘𝑃))))
54	53	3exp 1264	. . . . . 6 ⊢ (𝐷 ∈ V → (𝑦 ⊆ 𝐷 → (𝑦 ≈ 2𝑜 → ((𝑇‘𝑦) = 𝐹 → ((𝐷 ∈ V ∧ 𝑃 ⊆ 𝐷 ∧ 𝑃 ≈ 2𝑜) ∧ 𝐹 = (𝑇‘𝑃))))))
55	54	imp4a 614	. . . . 5 ⊢ (𝐷 ∈ V → (𝑦 ⊆ 𝐷 → ((𝑦 ≈ 2𝑜 ∧ (𝑇‘𝑦) = 𝐹) → ((𝐷 ∈ V ∧ 𝑃 ⊆ 𝐷 ∧ 𝑃 ≈ 2𝑜) ∧ 𝐹 = (𝑇‘𝑃)))))
56	28, 55	syl5 34	. . . 4 ⊢ (𝐷 ∈ V → (𝑦 ∈ 𝒫 𝐷 → ((𝑦 ≈ 2𝑜 ∧ (𝑇‘𝑦) = 𝐹) → ((𝐷 ∈ V ∧ 𝑃 ⊆ 𝐷 ∧ 𝑃 ≈ 2𝑜) ∧ 𝐹 = (𝑇‘𝑃)))))
57	56	rexlimdv 3030	. . 3 ⊢ (𝐷 ∈ V → (∃𝑦 ∈ 𝒫 𝐷(𝑦 ≈ 2𝑜 ∧ (𝑇‘𝑦) = 𝐹) → ((𝐷 ∈ V ∧ 𝑃 ⊆ 𝐷 ∧ 𝑃 ≈ 2𝑜) ∧ 𝐹 = (𝑇‘𝑃))))
58	27, 57	sylbid 230	. 2 ⊢ (𝐷 ∈ V → (𝐹 ∈ 𝑅 → ((𝐷 ∈ V ∧ 𝑃 ⊆ 𝐷 ∧ 𝑃 ≈ 2𝑜) ∧ 𝐹 = (𝑇‘𝑃))))
59	12, 58	mpcom 38	1 ⊢ (𝐹 ∈ 𝑅 → ((𝐷 ∈ V ∧ 𝑃 ⊆ 𝐷 ∧ 𝑃 ≈ 2𝑜) ∧ 𝐹 = (𝑇‘𝑃)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 196   ∧ wa 384   ∧ w3a 1037   = wceq 1483   ∈ wcel 1990  ∀wral 2912  ∃wrex 2913  {crab 2916  Vcvv 3200   ∖ cdif 3571   ⊆ wss 3574  ∅c0 3915  ifcif 4086  𝒫 cpw 4158  {csn 4177  ∪ cuni 4436   class class class wbr 4653   ↦ cmpt 4729   I cid 5023  dom cdm 5114  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-8 1992  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  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  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-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  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-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  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-om 7066  df-1o 7560  df-2o 7561  df-er 7742  df-en 7956  df-fin 7959  df-pmtr 17862

This theorem is referenced by:  pmtrffv  17879  pmtrrn2  17880  pmtrfinv  17881  pmtrfmvdn0  17882  pmtrff1o  17883  pmtrfcnv  17884  pmtrfb  17885