Theorem pmtrfrn 19652

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 4284	. . . 4 ⊢ ¬ 𝐹 ∈ ∅
2		pmtrrn.r	. . . . . 6 ⊢ 𝑅 = ran 𝑇
3		pmtrrn.t	. . . . . . 7 ⊢ 𝑇 = (pmTrsp‘𝐷)
4	3	rnfvprc 6871	. . . . . 6 ⊢ (¬ 𝐷 ∈ V → ran 𝑇 = ∅)
5	2, 4	eqtrid 2808	. . . . 5 ⊢ (¬ 𝐷 ∈ V → 𝑅 = ∅)
6	5	eleq2d 2847	. . . 4 ⊢ (¬ 𝐷 ∈ V → (𝐹 ∈ 𝑅 ↔ 𝐹 ∈ ∅))
7	1, 6	mtbiri 330	. . 3 ⊢ (¬ 𝐷 ∈ V → ¬ 𝐹 ∈ 𝑅)
8	7	con4i 115	. 2 ⊢ (𝐹 ∈ 𝑅 → 𝐷 ∈ V)
9		mptexg 7219	. . . . . . . 8 ⊢ (𝐷 ∈ V → (𝑧 ∈ 𝐷 ↦ if(𝑧 ∈ 𝑤, ∪ (𝑤 ∖ {𝑧}), 𝑧)) ∈ V)
10	9	ralrimivw 3159	. . . . . . 7 ⊢ (𝐷 ∈ V → ∀𝑤 ∈ {𝑥 ∈ 𝒫 𝐷 ∣ 𝑥 ≈ 2o} (𝑧 ∈ 𝐷 ↦ if(𝑧 ∈ 𝑤, ∪ (𝑤 ∖ {𝑧}), 𝑧)) ∈ V)
11		eqid 2761	. . . . . . . 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)))
12	11	fnmpt 6671	. . . . . . 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})
13	10, 12	syl 18	. . . . . 6 ⊢ (𝐷 ∈ V → (𝑤 ∈ {𝑥 ∈ 𝒫 𝐷 ∣ 𝑥 ≈ 2o} ↦ (𝑧 ∈ 𝐷 ↦ if(𝑧 ∈ 𝑤, ∪ (𝑤 ∖ {𝑧}), 𝑧))) Fn {𝑥 ∈ 𝒫 𝐷 ∣ 𝑥 ≈ 2o})
14	3	pmtrfval 19644	. . . . . . 7 ⊢ (𝐷 ∈ V → 𝑇 = (𝑤 ∈ {𝑥 ∈ 𝒫 𝐷 ∣ 𝑥 ≈ 2o} ↦ (𝑧 ∈ 𝐷 ↦ if(𝑧 ∈ 𝑤, ∪ (𝑤 ∖ {𝑧}), 𝑧))))
15	14	fneq1d 6624	. . . . . 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}))
16	13, 15	mpbird 260	. . . . 5 ⊢ (𝐷 ∈ V → 𝑇 Fn {𝑥 ∈ 𝒫 𝐷 ∣ 𝑥 ≈ 2o})
17		fvelrnb 6937	. . . . 5 ⊢ (𝑇 Fn {𝑥 ∈ 𝒫 𝐷 ∣ 𝑥 ≈ 2o} → (𝐹 ∈ ran 𝑇 ↔ ∃𝑦 ∈ {𝑥 ∈ 𝒫 𝐷 ∣ 𝑥 ≈ 2o} (𝑇‘𝑦) = 𝐹))
18	16, 17	syl 18	. . . 4 ⊢ (𝐷 ∈ V → (𝐹 ∈ ran 𝑇 ↔ ∃𝑦 ∈ {𝑥 ∈ 𝒫 𝐷 ∣ 𝑥 ≈ 2o} (𝑇‘𝑦) = 𝐹))
19	2	eleq2i 2853	. . . 4 ⊢ (𝐹 ∈ 𝑅 ↔ 𝐹 ∈ ran 𝑇)
20		breq1 5106	. . . . . 6 ⊢ (𝑥 = 𝑦 → (𝑥 ≈ 2o ↔ 𝑦 ≈ 2o))
21	20	rexrab 3654	. . . . 5 ⊢ (∃𝑦 ∈ {𝑥 ∈ 𝒫 𝐷 ∣ 𝑥 ≈ 2o} (𝑇‘𝑦) = 𝐹 ↔ ∃𝑦 ∈ 𝒫 𝐷(𝑦 ≈ 2o ∧ (𝑇‘𝑦) = 𝐹))
22	21	bicomi 227	. . . 4 ⊢ (∃𝑦 ∈ 𝒫 𝐷(𝑦 ≈ 2o ∧ (𝑇‘𝑦) = 𝐹) ↔ ∃𝑦 ∈ {𝑥 ∈ 𝒫 𝐷 ∣ 𝑥 ≈ 2o} (𝑇‘𝑦) = 𝐹)
23	18, 19, 22	3bitr4g 317	. . 3 ⊢ (𝐷 ∈ V → (𝐹 ∈ 𝑅 ↔ ∃𝑦 ∈ 𝒫 𝐷(𝑦 ≈ 2o ∧ (𝑇‘𝑦) = 𝐹)))
24		elpwi 4564	. . . . 5 ⊢ (𝑦 ∈ 𝒫 𝐷 → 𝑦 ⊆ 𝐷)
25		simp1 1154	. . . . . . . . . 10 ⊢ ((𝐷 ∈ V ∧ 𝑦 ⊆ 𝐷 ∧ 𝑦 ≈ 2o) → 𝐷 ∈ V)
26	3	pmtrmvd 19650	. . . . . . . . . . 11 ⊢ ((𝐷 ∈ V ∧ 𝑦 ⊆ 𝐷 ∧ 𝑦 ≈ 2o) → dom ((𝑇‘𝑦) ∖ I ) = 𝑦)
27		simp2 1155	. . . . . . . . . . 11 ⊢ ((𝐷 ∈ V ∧ 𝑦 ⊆ 𝐷 ∧ 𝑦 ≈ 2o) → 𝑦 ⊆ 𝐷)
28	26, 27	eqsstrd 3965	. . . . . . . . . 10 ⊢ ((𝐷 ∈ V ∧ 𝑦 ⊆ 𝐷 ∧ 𝑦 ≈ 2o) → dom ((𝑇‘𝑦) ∖ I ) ⊆ 𝐷)
29		simp3 1156	. . . . . . . . . . 11 ⊢ ((𝐷 ∈ V ∧ 𝑦 ⊆ 𝐷 ∧ 𝑦 ≈ 2o) → 𝑦 ≈ 2o)
30	26, 29	eqbrtrd 5127	. . . . . . . . . 10 ⊢ ((𝐷 ∈ V ∧ 𝑦 ⊆ 𝐷 ∧ 𝑦 ≈ 2o) → dom ((𝑇‘𝑦) ∖ I ) ≈ 2o)
31	25, 28, 30	3jca 1146	. . . . . . . . 9 ⊢ ((𝐷 ∈ V ∧ 𝑦 ⊆ 𝐷 ∧ 𝑦 ≈ 2o) → (𝐷 ∈ V ∧ dom ((𝑇‘𝑦) ∖ I ) ⊆ 𝐷 ∧ dom ((𝑇‘𝑦) ∖ I ) ≈ 2o))
32	26	eqcomd 2767	. . . . . . . . . 10 ⊢ ((𝐷 ∈ V ∧ 𝑦 ⊆ 𝐷 ∧ 𝑦 ≈ 2o) → 𝑦 = dom ((𝑇‘𝑦) ∖ I ))
33	32	fveq2d 6881	. . . . . . . . 9 ⊢ ((𝐷 ∈ V ∧ 𝑦 ⊆ 𝐷 ∧ 𝑦 ≈ 2o) → (𝑇‘𝑦) = (𝑇‘dom ((𝑇‘𝑦) ∖ I )))
34	31, 33	jca 521	. . . . . . . 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 ))))
35		difeq1 4067	. . . . . . . . . . 11 ⊢ ((𝑇‘𝑦) = 𝐹 → ((𝑇‘𝑦) ∖ I ) = (𝐹 ∖ I ))
36	35	dmeqd 5887	. . . . . . . . . 10 ⊢ ((𝑇‘𝑦) = 𝐹 → dom ((𝑇‘𝑦) ∖ I ) = dom (𝐹 ∖ I ))
37		pmtrfrn.p	. . . . . . . . . 10 ⊢ 𝑃 = dom (𝐹 ∖ I )
38	36, 37	eqtr4di 2814	. . . . . . . . 9 ⊢ ((𝑇‘𝑦) = 𝐹 → dom ((𝑇‘𝑦) ∖ I ) = 𝑃)
39		sseq1 3956	. . . . . . . . . . . 12 ⊢ (dom ((𝑇‘𝑦) ∖ I ) = 𝑃 → (dom ((𝑇‘𝑦) ∖ I ) ⊆ 𝐷 ↔ 𝑃 ⊆ 𝐷))
40		breq1 5106	. . . . . . . . . . . 12 ⊢ (dom ((𝑇‘𝑦) ∖ I ) = 𝑃 → (dom ((𝑇‘𝑦) ∖ I ) ≈ 2o ↔ 𝑃 ≈ 2o))
41	39, 40	3anbi23d 1467	. . . . . . . . . . 11 ⊢ (dom ((𝑇‘𝑦) ∖ I ) = 𝑃 → ((𝐷 ∈ V ∧ dom ((𝑇‘𝑦) ∖ I ) ⊆ 𝐷 ∧ dom ((𝑇‘𝑦) ∖ I ) ≈ 2o) ↔ (𝐷 ∈ V ∧ 𝑃 ⊆ 𝐷 ∧ 𝑃 ≈ 2o)))
42	41	adantl 487	. . . . . . . . . 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)))
43		simpl 488	. . . . . . . . . . 11 ⊢ (((𝑇‘𝑦) = 𝐹 ∧ dom ((𝑇‘𝑦) ∖ I ) = 𝑃) → (𝑇‘𝑦) = 𝐹)
44		fveq2 6877	. . . . . . . . . . . 12 ⊢ (dom ((𝑇‘𝑦) ∖ I ) = 𝑃 → (𝑇‘dom ((𝑇‘𝑦) ∖ I )) = (𝑇‘𝑃))
45	44	adantl 487	. . . . . . . . . . 11 ⊢ (((𝑇‘𝑦) = 𝐹 ∧ dom ((𝑇‘𝑦) ∖ I ) = 𝑃) → (𝑇‘dom ((𝑇‘𝑦) ∖ I )) = (𝑇‘𝑃))
46	43, 45	eqeq12d 2777	. . . . . . . . . 10 ⊢ (((𝑇‘𝑦) = 𝐹 ∧ dom ((𝑇‘𝑦) ∖ I ) = 𝑃) → ((𝑇‘𝑦) = (𝑇‘dom ((𝑇‘𝑦) ∖ I )) ↔ 𝐹 = (𝑇‘𝑃)))
47	42, 46	anbi12d 644	. . . . . . . . 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))))
48	38, 47	mpdan 700	. . . . . . . 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))))
49	34, 48	syl5ibcom 248	. . . . . . 7 ⊢ ((𝐷 ∈ V ∧ 𝑦 ⊆ 𝐷 ∧ 𝑦 ≈ 2o) → ((𝑇‘𝑦) = 𝐹 → ((𝐷 ∈ V ∧ 𝑃 ⊆ 𝐷 ∧ 𝑃 ≈ 2o) ∧ 𝐹 = (𝑇‘𝑃))))
50	49	3exp 1137	. . . . . 6 ⊢ (𝐷 ∈ V → (𝑦 ⊆ 𝐷 → (𝑦 ≈ 2o → ((𝑇‘𝑦) = 𝐹 → ((𝐷 ∈ V ∧ 𝑃 ⊆ 𝐷 ∧ 𝑃 ≈ 2o) ∧ 𝐹 = (𝑇‘𝑃))))))
51	50	imp4a 428	. . . . 5 ⊢ (𝐷 ∈ V → (𝑦 ⊆ 𝐷 → ((𝑦 ≈ 2o ∧ (𝑇‘𝑦) = 𝐹) → ((𝐷 ∈ V ∧ 𝑃 ⊆ 𝐷 ∧ 𝑃 ≈ 2o) ∧ 𝐹 = (𝑇‘𝑃)))))
52	24, 51	syl5 35	. . . 4 ⊢ (𝐷 ∈ V → (𝑦 ∈ 𝒫 𝐷 → ((𝑦 ≈ 2o ∧ (𝑇‘𝑦) = 𝐹) → ((𝐷 ∈ V ∧ 𝑃 ⊆ 𝐷 ∧ 𝑃 ≈ 2o) ∧ 𝐹 = (𝑇‘𝑃)))))
53	52	rexlimdv 3162	. . 3 ⊢ (𝐷 ∈ V → (∃𝑦 ∈ 𝒫 𝐷(𝑦 ≈ 2o ∧ (𝑇‘𝑦) = 𝐹) → ((𝐷 ∈ V ∧ 𝑃 ⊆ 𝐷 ∧ 𝑃 ≈ 2o) ∧ 𝐹 = (𝑇‘𝑃))))
54	23, 53	sylbid 243	. 2 ⊢ (𝐷 ∈ V → (𝐹 ∈ 𝑅 → ((𝐷 ∈ V ∧ 𝑃 ⊆ 𝐷 ∧ 𝑃 ≈ 2o) ∧ 𝐹 = (𝑇‘𝑃))))
55	8, 54	mpcom 39	1 ⊢ (𝐹 ∈ 𝑅 → ((𝐷 ∈ V ∧ 𝑃 ⊆ 𝐷 ∧ 𝑃 ≈ 2o) ∧ 𝐹 = (𝑇‘𝑃)))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2145  ∀wral 3077  ∃wrex 3087  {crab 3413  Vcvv 3451   ∖ cdif 3896   ⊆ wss 3899  ∅c0 4279  ifcif 4482  𝒫 cpw 4557  {csn 4584  ∪ cuni 4867   class class class wbr 5103   ↦ cmpt 5186   I cid 5545  dom cdm 5651  ran crn 5652   Fn wfn 6526  ‘cfv 6531  2_oc2o 8454   ≈ cen 8954  pmTrspcpmtr 19635

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-rep 5232  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391  ax-un 7740

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-reu 3367  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5546  df-eprel 5551  df-po 5559  df-so 5560  df-fr 5604  df-we 5606  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-ord 6358  df-on 6359  df-lim 6360  df-suc 6361  df-iota 6487  df-fun 6533  df-fn 6534  df-f 6535  df-f1 6536  df-fo 6537  df-f1o 6538  df-fv 6539  df-om 7867  df-1o 8460  df-2o 8461  df-en 8958  df-pmtr 19636

This theorem is used by:  pmtrffv  19653  pmtrrn2  19654  pmtrfinv  19655  pmtrfmvdn0  19656  pmtrff1o  19657  pmtrfcnv  19658  pmtrfb  19659