Theorem pmtrfrn 19439

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 4313	. . . 4 ⊢ ¬ 𝐹 ∈ ∅
2		pmtrrn.r	. . . . . 6 ⊢ 𝑅 = ran 𝑇
3		pmtrrn.t	. . . . . . 7 ⊢ 𝑇 = (pmTrsp‘𝐷)
4	3	rnfvprc 6870	. . . . . 6 ⊢ (¬ 𝐷 ∈ V → ran 𝑇 = ∅)
5	2, 4	eqtrid 2782	. . . . 5 ⊢ (¬ 𝐷 ∈ V → 𝑅 = ∅)
6	5	eleq2d 2820	. . . 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 7213	. . . . . . . 8 ⊢ (𝐷 ∈ V → (𝑧 ∈ 𝐷 ↦ if(𝑧 ∈ 𝑤, ∪ (𝑤 ∖ {𝑧}), 𝑧)) ∈ V)
10	9	ralrimivw 3136	. . . . . . 7 ⊢ (𝐷 ∈ V → ∀𝑤 ∈ {𝑥 ∈ 𝒫 𝐷 ∣ 𝑥 ≈ 2o} (𝑧 ∈ 𝐷 ↦ if(𝑧 ∈ 𝑤, ∪ (𝑤 ∖ {𝑧}), 𝑧)) ∈ V)
11		eqid 2735	. . . . . . . 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 6678	. . . . . . 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 19431	. . . . . . 7 ⊢ (𝐷 ∈ V → 𝑇 = (𝑤 ∈ {𝑥 ∈ 𝒫 𝐷 ∣ 𝑥 ≈ 2o} ↦ (𝑧 ∈ 𝐷 ↦ if(𝑧 ∈ 𝑤, ∪ (𝑤 ∖ {𝑧}), 𝑧))))
15	14	fneq1d 6631	. . . . . 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 6939	. . . . 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 2826	. . . 4 ⊢ (𝐹 ∈ 𝑅 ↔ 𝐹 ∈ ran 𝑇)
20		breq1 5122	. . . . . 6 ⊢ (𝑥 = 𝑦 → (𝑥 ≈ 2o ↔ 𝑦 ≈ 2o))
21	20	rexrab 3679	. . . . 5 ⊢ (∃𝑦 ∈ {𝑥 ∈ 𝒫 𝐷 ∣ 𝑥 ≈ 2o} (𝑇‘𝑦) = 𝐹 ↔ ∃𝑦 ∈ 𝒫 𝐷(𝑦 ≈ 2o ∧ (𝑇‘𝑦) = 𝐹))
22	21	bicomi 224	. . . 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 4582	. . . . 5 ⊢ (𝑦 ∈ 𝒫 𝐷 → 𝑦 ⊆ 𝐷)
25		simp1 1136	. . . . . . . . . 10 ⊢ ((𝐷 ∈ V ∧ 𝑦 ⊆ 𝐷 ∧ 𝑦 ≈ 2o) → 𝐷 ∈ V)
26	3	pmtrmvd 19437	. . . . . . . . . . 11 ⊢ ((𝐷 ∈ V ∧ 𝑦 ⊆ 𝐷 ∧ 𝑦 ≈ 2o) → dom ((𝑇‘𝑦) ∖ I ) = 𝑦)
27		simp2 1137	. . . . . . . . . . 11 ⊢ ((𝐷 ∈ V ∧ 𝑦 ⊆ 𝐷 ∧ 𝑦 ≈ 2o) → 𝑦 ⊆ 𝐷)
28	26, 27	eqsstrd 3993	. . . . . . . . . 10 ⊢ ((𝐷 ∈ V ∧ 𝑦 ⊆ 𝐷 ∧ 𝑦 ≈ 2o) → dom ((𝑇‘𝑦) ∖ I ) ⊆ 𝐷)
29		simp3 1138	. . . . . . . . . . 11 ⊢ ((𝐷 ∈ V ∧ 𝑦 ⊆ 𝐷 ∧ 𝑦 ≈ 2o) → 𝑦 ≈ 2o)
30	26, 29	eqbrtrd 5141	. . . . . . . . . 10 ⊢ ((𝐷 ∈ V ∧ 𝑦 ⊆ 𝐷 ∧ 𝑦 ≈ 2o) → dom ((𝑇‘𝑦) ∖ I ) ≈ 2o)
31	25, 28, 30	3jca 1128	. . . . . . . . 9 ⊢ ((𝐷 ∈ V ∧ 𝑦 ⊆ 𝐷 ∧ 𝑦 ≈ 2o) → (𝐷 ∈ V ∧ dom ((𝑇‘𝑦) ∖ I ) ⊆ 𝐷 ∧ dom ((𝑇‘𝑦) ∖ I ) ≈ 2o))
32	26	eqcomd 2741	. . . . . . . . . 10 ⊢ ((𝐷 ∈ V ∧ 𝑦 ⊆ 𝐷 ∧ 𝑦 ≈ 2o) → 𝑦 = dom ((𝑇‘𝑦) ∖ I ))
33	32	fveq2d 6880	. . . . . . . . 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 4094	. . . . . . . . . . 11 ⊢ ((𝑇‘𝑦) = 𝐹 → ((𝑇‘𝑦) ∖ I ) = (𝐹 ∖ I ))
36	35	dmeqd 5885	. . . . . . . . . 10 ⊢ ((𝑇‘𝑦) = 𝐹 → dom ((𝑇‘𝑦) ∖ I ) = dom (𝐹 ∖ I ))
37		pmtrfrn.p	. . . . . . . . . 10 ⊢ 𝑃 = dom (𝐹 ∖ I )
38	36, 37	eqtr4di 2788	. . . . . . . . 9 ⊢ ((𝑇‘𝑦) = 𝐹 → dom ((𝑇‘𝑦) ∖ I ) = 𝑃)
39		sseq1 3984	. . . . . . . . . . . 12 ⊢ (dom ((𝑇‘𝑦) ∖ I ) = 𝑃 → (dom ((𝑇‘𝑦) ∖ I ) ⊆ 𝐷 ↔ 𝑃 ⊆ 𝐷))
40		breq1 5122	. . . . . . . . . . . 12 ⊢ (dom ((𝑇‘𝑦) ∖ I ) = 𝑃 → (dom ((𝑇‘𝑦) ∖ I ) ≈ 2o ↔ 𝑃 ≈ 2o))
41	39, 40	3anbi23d 1441	. . . . . . . . . . 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 6876	. . . . . . . . . . . 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 2751	. . . . . . . . . 10 ⊢ (((𝑇‘𝑦) = 𝐹 ∧ dom ((𝑇‘𝑦) ∖ I ) = 𝑃) → ((𝑇‘𝑦) = (𝑇‘dom ((𝑇‘𝑦) ∖ I )) ↔ 𝐹 = (𝑇‘𝑃)))
47	42, 46	anbi12d 632	. . . . . . . . 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 687	. . . . . . . 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 245	. . . . . . 7 ⊢ ((𝐷 ∈ V ∧ 𝑦 ⊆ 𝐷 ∧ 𝑦 ≈ 2o) → ((𝑇‘𝑦) = 𝐹 → ((𝐷 ∈ V ∧ 𝑃 ⊆ 𝐷 ∧ 𝑃 ≈ 2o) ∧ 𝐹 = (𝑇‘𝑃))))
50	49	3exp 1119	. . . . . 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 3139	. . 3 ⊢ (𝐷 ∈ V → (∃𝑦 ∈ 𝒫 𝐷(𝑦 ≈ 2o ∧ (𝑇‘𝑦) = 𝐹) → ((𝐷 ∈ V ∧ 𝑃 ⊆ 𝐷 ∧ 𝑃 ≈ 2o) ∧ 𝐹 = (𝑇‘𝑃))))
54	23, 53	sylbid 240	. 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 206   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2108  ∀wral 3051  ∃wrex 3060  {crab 3415  Vcvv 3459   ∖ cdif 3923   ⊆ wss 3926  ∅c0 4308  ifcif 4500  𝒫 cpw 4575  {csn 4601  ∪ cuni 4883   class class class wbr 5119   ↦ cmpt 5201   I cid 5547  dom cdm 5654  ran crn 5655   Fn wfn 6526  ‘cfv 6531  2_oc2o 8474   ≈ cen 8956  pmTrspcpmtr 19422

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2707  ax-rep 5249  ax-sep 5266  ax-nul 5276  ax-pow 5335  ax-pr 5402  ax-un 7729

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2809  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-reu 3360  df-rab 3416  df-v 3461  df-sbc 3766  df-csb 3875  df-dif 3929  df-un 3931  df-in 3933  df-ss 3943  df-pss 3946  df-nul 4309  df-if 4501  df-pw 4577  df-sn 4602  df-pr 4604  df-op 4608  df-uni 4884  df-iun 4969  df-br 5120  df-opab 5182  df-mpt 5202  df-tr 5230  df-id 5548  df-eprel 5553  df-po 5561  df-so 5562  df-fr 5606  df-we 5608  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-rn 5665  df-res 5666  df-ima 5667  df-ord 6355  df-on 6356  df-lim 6357  df-suc 6358  df-iota 6484  df-fun 6533  df-fn 6534  df-f 6535  df-f1 6536  df-fo 6537  df-f1o 6538  df-fv 6539  df-om 7862  df-1o 8480  df-2o 8481  df-en 8960  df-pmtr 19423

This theorem is referenced by:  pmtrffv  19440  pmtrrn2  19441  pmtrfinv  19442  pmtrfmvdn0  19443  pmtrff1o  19444  pmtrfcnv  19445  pmtrfb  19446