Theorem pmtrfrn 19375

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 4325	. . . 4 ⊢ ¬ 𝐹 ∈ ∅
2		pmtrrn.r	. . . . . 6 ⊢ 𝑅 = ran 𝑇
3		pmtrrn.t	. . . . . . 7 ⊢ 𝑇 = (pmTrsp‘𝐷)
4	3	rnfvprc 6878	. . . . . 6 ⊢ (¬ 𝐷 ∈ V → ran 𝑇 = ∅)
5	2, 4	eqtrid 2778	. . . . 5 ⊢ (¬ 𝐷 ∈ V → 𝑅 = ∅)
6	5	eleq2d 2813	. . . 4 ⊢ (¬ 𝐷 ∈ V → (𝐹 ∈ 𝑅 ↔ 𝐹 ∈ ∅))
7	1, 6	mtbiri 327	. . 3 ⊢ (¬ 𝐷 ∈ V → ¬ 𝐹 ∈ 𝑅)
8	7	con4i 114	. 2 ⊢ (𝐹 ∈ 𝑅 → 𝐷 ∈ V)
9		mptexg 7217	. . . . . . . 8 ⊢ (𝐷 ∈ V → (𝑧 ∈ 𝐷 ↦ if(𝑧 ∈ 𝑤, ∪ (𝑤 ∖ {𝑧}), 𝑧)) ∈ V)
10	9	ralrimivw 3144	. . . . . . 7 ⊢ (𝐷 ∈ V → ∀𝑤 ∈ {𝑥 ∈ 𝒫 𝐷 ∣ 𝑥 ≈ 2o} (𝑧 ∈ 𝐷 ↦ if(𝑧 ∈ 𝑤, ∪ (𝑤 ∖ {𝑧}), 𝑧)) ∈ V)
11		eqid 2726	. . . . . . . 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 6683	. . . . . . 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 17	. . . . . 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 19367	. . . . . . 7 ⊢ (𝐷 ∈ V → 𝑇 = (𝑤 ∈ {𝑥 ∈ 𝒫 𝐷 ∣ 𝑥 ≈ 2o} ↦ (𝑧 ∈ 𝐷 ↦ if(𝑧 ∈ 𝑤, ∪ (𝑤 ∖ {𝑧}), 𝑧))))
15	14	fneq1d 6635	. . . . . 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 257	. . . . 5 ⊢ (𝐷 ∈ V → 𝑇 Fn {𝑥 ∈ 𝒫 𝐷 ∣ 𝑥 ≈ 2o})
17		fvelrnb 6945	. . . . 5 ⊢ (𝑇 Fn {𝑥 ∈ 𝒫 𝐷 ∣ 𝑥 ≈ 2o} → (𝐹 ∈ ran 𝑇 ↔ ∃𝑦 ∈ {𝑥 ∈ 𝒫 𝐷 ∣ 𝑥 ≈ 2o} (𝑇‘𝑦) = 𝐹))
18	16, 17	syl 17	. . . 4 ⊢ (𝐷 ∈ V → (𝐹 ∈ ran 𝑇 ↔ ∃𝑦 ∈ {𝑥 ∈ 𝒫 𝐷 ∣ 𝑥 ≈ 2o} (𝑇‘𝑦) = 𝐹))
19	2	eleq2i 2819	. . . 4 ⊢ (𝐹 ∈ 𝑅 ↔ 𝐹 ∈ ran 𝑇)
20		breq1 5144	. . . . . 6 ⊢ (𝑥 = 𝑦 → (𝑥 ≈ 2o ↔ 𝑦 ≈ 2o))
21	20	rexrab 3687	. . . . 5 ⊢ (∃𝑦 ∈ {𝑥 ∈ 𝒫 𝐷 ∣ 𝑥 ≈ 2o} (𝑇‘𝑦) = 𝐹 ↔ ∃𝑦 ∈ 𝒫 𝐷(𝑦 ≈ 2o ∧ (𝑇‘𝑦) = 𝐹))
22	21	bicomi 223	. . . 4 ⊢ (∃𝑦 ∈ 𝒫 𝐷(𝑦 ≈ 2o ∧ (𝑇‘𝑦) = 𝐹) ↔ ∃𝑦 ∈ {𝑥 ∈ 𝒫 𝐷 ∣ 𝑥 ≈ 2o} (𝑇‘𝑦) = 𝐹)
23	18, 19, 22	3bitr4g 314	. . 3 ⊢ (𝐷 ∈ V → (𝐹 ∈ 𝑅 ↔ ∃𝑦 ∈ 𝒫 𝐷(𝑦 ≈ 2o ∧ (𝑇‘𝑦) = 𝐹)))
24		elpwi 4604	. . . . 5 ⊢ (𝑦 ∈ 𝒫 𝐷 → 𝑦 ⊆ 𝐷)
25		simp1 1133	. . . . . . . . . 10 ⊢ ((𝐷 ∈ V ∧ 𝑦 ⊆ 𝐷 ∧ 𝑦 ≈ 2o) → 𝐷 ∈ V)
26	3	pmtrmvd 19373	. . . . . . . . . . 11 ⊢ ((𝐷 ∈ V ∧ 𝑦 ⊆ 𝐷 ∧ 𝑦 ≈ 2o) → dom ((𝑇‘𝑦) ∖ I ) = 𝑦)
27		simp2 1134	. . . . . . . . . . 11 ⊢ ((𝐷 ∈ V ∧ 𝑦 ⊆ 𝐷 ∧ 𝑦 ≈ 2o) → 𝑦 ⊆ 𝐷)
28	26, 27	eqsstrd 4015	. . . . . . . . . 10 ⊢ ((𝐷 ∈ V ∧ 𝑦 ⊆ 𝐷 ∧ 𝑦 ≈ 2o) → dom ((𝑇‘𝑦) ∖ I ) ⊆ 𝐷)
29		simp3 1135	. . . . . . . . . . 11 ⊢ ((𝐷 ∈ V ∧ 𝑦 ⊆ 𝐷 ∧ 𝑦 ≈ 2o) → 𝑦 ≈ 2o)
30	26, 29	eqbrtrd 5163	. . . . . . . . . 10 ⊢ ((𝐷 ∈ V ∧ 𝑦 ⊆ 𝐷 ∧ 𝑦 ≈ 2o) → dom ((𝑇‘𝑦) ∖ I ) ≈ 2o)
31	25, 28, 30	3jca 1125	. . . . . . . . 9 ⊢ ((𝐷 ∈ V ∧ 𝑦 ⊆ 𝐷 ∧ 𝑦 ≈ 2o) → (𝐷 ∈ V ∧ dom ((𝑇‘𝑦) ∖ I ) ⊆ 𝐷 ∧ dom ((𝑇‘𝑦) ∖ I ) ≈ 2o))
32	26	eqcomd 2732	. . . . . . . . . 10 ⊢ ((𝐷 ∈ V ∧ 𝑦 ⊆ 𝐷 ∧ 𝑦 ≈ 2o) → 𝑦 = dom ((𝑇‘𝑦) ∖ I ))
33	32	fveq2d 6888	. . . . . . . . 9 ⊢ ((𝐷 ∈ V ∧ 𝑦 ⊆ 𝐷 ∧ 𝑦 ≈ 2o) → (𝑇‘𝑦) = (𝑇‘dom ((𝑇‘𝑦) ∖ I )))
34	31, 33	jca 511	. . . . . . . 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 4110	. . . . . . . . . . 11 ⊢ ((𝑇‘𝑦) = 𝐹 → ((𝑇‘𝑦) ∖ I ) = (𝐹 ∖ I ))
36	35	dmeqd 5898	. . . . . . . . . 10 ⊢ ((𝑇‘𝑦) = 𝐹 → dom ((𝑇‘𝑦) ∖ I ) = dom (𝐹 ∖ I ))
37		pmtrfrn.p	. . . . . . . . . 10 ⊢ 𝑃 = dom (𝐹 ∖ I )
38	36, 37	eqtr4di 2784	. . . . . . . . 9 ⊢ ((𝑇‘𝑦) = 𝐹 → dom ((𝑇‘𝑦) ∖ I ) = 𝑃)
39		sseq1 4002	. . . . . . . . . . . 12 ⊢ (dom ((𝑇‘𝑦) ∖ I ) = 𝑃 → (dom ((𝑇‘𝑦) ∖ I ) ⊆ 𝐷 ↔ 𝑃 ⊆ 𝐷))
40		breq1 5144	. . . . . . . . . . . 12 ⊢ (dom ((𝑇‘𝑦) ∖ I ) = 𝑃 → (dom ((𝑇‘𝑦) ∖ I ) ≈ 2o ↔ 𝑃 ≈ 2o))
41	39, 40	3anbi23d 1435	. . . . . . . . . . 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 481	. . . . . . . . . 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 482	. . . . . . . . . . 11 ⊢ (((𝑇‘𝑦) = 𝐹 ∧ dom ((𝑇‘𝑦) ∖ I ) = 𝑃) → (𝑇‘𝑦) = 𝐹)
44		fveq2 6884	. . . . . . . . . . . 12 ⊢ (dom ((𝑇‘𝑦) ∖ I ) = 𝑃 → (𝑇‘dom ((𝑇‘𝑦) ∖ I )) = (𝑇‘𝑃))
45	44	adantl 481	. . . . . . . . . . 11 ⊢ (((𝑇‘𝑦) = 𝐹 ∧ dom ((𝑇‘𝑦) ∖ I ) = 𝑃) → (𝑇‘dom ((𝑇‘𝑦) ∖ I )) = (𝑇‘𝑃))
46	43, 45	eqeq12d 2742	. . . . . . . . . 10 ⊢ (((𝑇‘𝑦) = 𝐹 ∧ dom ((𝑇‘𝑦) ∖ I ) = 𝑃) → ((𝑇‘𝑦) = (𝑇‘dom ((𝑇‘𝑦) ∖ I )) ↔ 𝐹 = (𝑇‘𝑃)))
47	42, 46	anbi12d 630	. . . . . . . . 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 684	. . . . . . . 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 244	. . . . . . 7 ⊢ ((𝐷 ∈ V ∧ 𝑦 ⊆ 𝐷 ∧ 𝑦 ≈ 2o) → ((𝑇‘𝑦) = 𝐹 → ((𝐷 ∈ V ∧ 𝑃 ⊆ 𝐷 ∧ 𝑃 ≈ 2o) ∧ 𝐹 = (𝑇‘𝑃))))
50	49	3exp 1116	. . . . . 6 ⊢ (𝐷 ∈ V → (𝑦 ⊆ 𝐷 → (𝑦 ≈ 2o → ((𝑇‘𝑦) = 𝐹 → ((𝐷 ∈ V ∧ 𝑃 ⊆ 𝐷 ∧ 𝑃 ≈ 2o) ∧ 𝐹 = (𝑇‘𝑃))))))
51	50	imp4a 422	. . . . 5 ⊢ (𝐷 ∈ V → (𝑦 ⊆ 𝐷 → ((𝑦 ≈ 2o ∧ (𝑇‘𝑦) = 𝐹) → ((𝐷 ∈ V ∧ 𝑃 ⊆ 𝐷 ∧ 𝑃 ≈ 2o) ∧ 𝐹 = (𝑇‘𝑃)))))
52	24, 51	syl5 34	. . . 4 ⊢ (𝐷 ∈ V → (𝑦 ∈ 𝒫 𝐷 → ((𝑦 ≈ 2o ∧ (𝑇‘𝑦) = 𝐹) → ((𝐷 ∈ V ∧ 𝑃 ⊆ 𝐷 ∧ 𝑃 ≈ 2o) ∧ 𝐹 = (𝑇‘𝑃)))))
53	52	rexlimdv 3147	. . 3 ⊢ (𝐷 ∈ V → (∃𝑦 ∈ 𝒫 𝐷(𝑦 ≈ 2o ∧ (𝑇‘𝑦) = 𝐹) → ((𝐷 ∈ V ∧ 𝑃 ⊆ 𝐷 ∧ 𝑃 ≈ 2o) ∧ 𝐹 = (𝑇‘𝑃))))
54	23, 53	sylbid 239	. 2 ⊢ (𝐷 ∈ V → (𝐹 ∈ 𝑅 → ((𝐷 ∈ V ∧ 𝑃 ⊆ 𝐷 ∧ 𝑃 ≈ 2o) ∧ 𝐹 = (𝑇‘𝑃))))
55	8, 54	mpcom 38	1 ⊢ (𝐹 ∈ 𝑅 → ((𝐷 ∈ V ∧ 𝑃 ⊆ 𝐷 ∧ 𝑃 ≈ 2o) ∧ 𝐹 = (𝑇‘𝑃)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 395   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098  ∀wral 3055  ∃wrex 3064  {crab 3426  Vcvv 3468   ∖ cdif 3940   ⊆ wss 3943  ∅c0 4317  ifcif 4523  𝒫 cpw 4597  {csn 4623  ∪ cuni 4902   class class class wbr 5141   ↦ cmpt 5224   I cid 5566  dom cdm 5669  ran crn 5670   Fn wfn 6531  ‘cfv 6536  2_oc2o 8458   ≈ cen 8935  pmTrspcpmtr 19358

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7721

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-pss 3962  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-tr 5259  df-id 5567  df-eprel 5573  df-po 5581  df-so 5582  df-fr 5624  df-we 5626  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-ord 6360  df-on 6361  df-lim 6362  df-suc 6363  df-iota 6488  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-om 7852  df-1o 8464  df-2o 8465  df-en 8939  df-pmtr 19359

This theorem is referenced by:  pmtrffv  19376  pmtrrn2  19377  pmtrfinv  19378  pmtrfmvdn0  19379  pmtrff1o  19380  pmtrfcnv  19381  pmtrfb  19382