Theorem pmtrfrn 19399

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 4292	. . . 4 ⊢ ¬ 𝐹 ∈ ∅
2		pmtrrn.r	. . . . . 6 ⊢ 𝑅 = ran 𝑇
3		pmtrrn.t	. . . . . . 7 ⊢ 𝑇 = (pmTrsp‘𝐷)
4	3	rnfvprc 6836	. . . . . 6 ⊢ (¬ 𝐷 ∈ V → ran 𝑇 = ∅)
5	2, 4	eqtrid 2784	. . . . 5 ⊢ (¬ 𝐷 ∈ V → 𝑅 = ∅)
6	5	eleq2d 2823	. . . 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 7177	. . . . . . . 8 ⊢ (𝐷 ∈ V → (𝑧 ∈ 𝐷 ↦ if(𝑧 ∈ 𝑤, ∪ (𝑤 ∖ {𝑧}), 𝑧)) ∈ V)
10	9	ralrimivw 3134	. . . . . . 7 ⊢ (𝐷 ∈ V → ∀𝑤 ∈ {𝑥 ∈ 𝒫 𝐷 ∣ 𝑥 ≈ 2o} (𝑧 ∈ 𝐷 ↦ if(𝑧 ∈ 𝑤, ∪ (𝑤 ∖ {𝑧}), 𝑧)) ∈ V)
11		eqid 2737	. . . . . . . 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 6640	. . . . . . 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 19391	. . . . . . 7 ⊢ (𝐷 ∈ V → 𝑇 = (𝑤 ∈ {𝑥 ∈ 𝒫 𝐷 ∣ 𝑥 ≈ 2o} ↦ (𝑧 ∈ 𝐷 ↦ if(𝑧 ∈ 𝑤, ∪ (𝑤 ∖ {𝑧}), 𝑧))))
15	14	fneq1d 6593	. . . . . 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 6902	. . . . 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 2829	. . . 4 ⊢ (𝐹 ∈ 𝑅 ↔ 𝐹 ∈ ran 𝑇)
20		breq1 5103	. . . . . 6 ⊢ (𝑥 = 𝑦 → (𝑥 ≈ 2o ↔ 𝑦 ≈ 2o))
21	20	rexrab 3656	. . . . 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 4563	. . . . 5 ⊢ (𝑦 ∈ 𝒫 𝐷 → 𝑦 ⊆ 𝐷)
25		simp1 1137	. . . . . . . . . 10 ⊢ ((𝐷 ∈ V ∧ 𝑦 ⊆ 𝐷 ∧ 𝑦 ≈ 2o) → 𝐷 ∈ V)
26	3	pmtrmvd 19397	. . . . . . . . . . 11 ⊢ ((𝐷 ∈ V ∧ 𝑦 ⊆ 𝐷 ∧ 𝑦 ≈ 2o) → dom ((𝑇‘𝑦) ∖ I ) = 𝑦)
27		simp2 1138	. . . . . . . . . . 11 ⊢ ((𝐷 ∈ V ∧ 𝑦 ⊆ 𝐷 ∧ 𝑦 ≈ 2o) → 𝑦 ⊆ 𝐷)
28	26, 27	eqsstrd 3970	. . . . . . . . . 10 ⊢ ((𝐷 ∈ V ∧ 𝑦 ⊆ 𝐷 ∧ 𝑦 ≈ 2o) → dom ((𝑇‘𝑦) ∖ I ) ⊆ 𝐷)
29		simp3 1139	. . . . . . . . . . 11 ⊢ ((𝐷 ∈ V ∧ 𝑦 ⊆ 𝐷 ∧ 𝑦 ≈ 2o) → 𝑦 ≈ 2o)
30	26, 29	eqbrtrd 5122	. . . . . . . . . 10 ⊢ ((𝐷 ∈ V ∧ 𝑦 ⊆ 𝐷 ∧ 𝑦 ≈ 2o) → dom ((𝑇‘𝑦) ∖ I ) ≈ 2o)
31	25, 28, 30	3jca 1129	. . . . . . . . 9 ⊢ ((𝐷 ∈ V ∧ 𝑦 ⊆ 𝐷 ∧ 𝑦 ≈ 2o) → (𝐷 ∈ V ∧ dom ((𝑇‘𝑦) ∖ I ) ⊆ 𝐷 ∧ dom ((𝑇‘𝑦) ∖ I ) ≈ 2o))
32	26	eqcomd 2743	. . . . . . . . . 10 ⊢ ((𝐷 ∈ V ∧ 𝑦 ⊆ 𝐷 ∧ 𝑦 ≈ 2o) → 𝑦 = dom ((𝑇‘𝑦) ∖ I ))
33	32	fveq2d 6846	. . . . . . . . 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 4073	. . . . . . . . . . 11 ⊢ ((𝑇‘𝑦) = 𝐹 → ((𝑇‘𝑦) ∖ I ) = (𝐹 ∖ I ))
36	35	dmeqd 5862	. . . . . . . . . 10 ⊢ ((𝑇‘𝑦) = 𝐹 → dom ((𝑇‘𝑦) ∖ I ) = dom (𝐹 ∖ I ))
37		pmtrfrn.p	. . . . . . . . . 10 ⊢ 𝑃 = dom (𝐹 ∖ I )
38	36, 37	eqtr4di 2790	. . . . . . . . 9 ⊢ ((𝑇‘𝑦) = 𝐹 → dom ((𝑇‘𝑦) ∖ I ) = 𝑃)
39		sseq1 3961	. . . . . . . . . . . 12 ⊢ (dom ((𝑇‘𝑦) ∖ I ) = 𝑃 → (dom ((𝑇‘𝑦) ∖ I ) ⊆ 𝐷 ↔ 𝑃 ⊆ 𝐷))
40		breq1 5103	. . . . . . . . . . . 12 ⊢ (dom ((𝑇‘𝑦) ∖ I ) = 𝑃 → (dom ((𝑇‘𝑦) ∖ I ) ≈ 2o ↔ 𝑃 ≈ 2o))
41	39, 40	3anbi23d 1442	. . . . . . . . . . 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 6842	. . . . . . . . . . . 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 2753	. . . . . . . . . 10 ⊢ (((𝑇‘𝑦) = 𝐹 ∧ dom ((𝑇‘𝑦) ∖ I ) = 𝑃) → ((𝑇‘𝑦) = (𝑇‘dom ((𝑇‘𝑦) ∖ I )) ↔ 𝐹 = (𝑇‘𝑃)))
47	42, 46	anbi12d 633	. . . . . . . . 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 688	. . . . . . . 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 1120	. . . . . 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 3137	. . 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 1087   = wceq 1542   ∈ wcel 2114  ∀wral 3052  ∃wrex 3062  {crab 3401  Vcvv 3442   ∖ cdif 3900   ⊆ wss 3903  ∅c0 4287  ifcif 4481  𝒫 cpw 4556  {csn 4582  ∪ cuni 4865   class class class wbr 5100   ↦ cmpt 5181   I cid 5526  dom cdm 5632  ran crn 5633   Fn wfn 6495  ‘cfv 6500  2_oc2o 8401   ≈ cen 8892  pmTrspcpmtr 19382

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5226  ax-sep 5243  ax-nul 5253  ax-pow 5312  ax-pr 5379  ax-un 7690

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-reu 3353  df-rab 3402  df-v 3444  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-iun 4950  df-br 5101  df-opab 5163  df-mpt 5182  df-tr 5208  df-id 5527  df-eprel 5532  df-po 5540  df-so 5541  df-fr 5585  df-we 5587  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-ord 6328  df-on 6329  df-lim 6330  df-suc 6331  df-iota 6456  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-om 7819  df-1o 8407  df-2o 8408  df-en 8896  df-pmtr 19383

This theorem is referenced by:  pmtrffv  19400  pmtrrn2  19401  pmtrfinv  19402  pmtrfmvdn0  19403  pmtrff1o  19404  pmtrfcnv  19405  pmtrfb  19406