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Theorem pmtrfrn 17924
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 (𝐹𝑅 → ((𝐷 ∈ V ∧ 𝑃𝐷𝑃 ≈ 2𝑜) ∧ 𝐹 = (𝑇𝑃)))

Proof of Theorem pmtrfrn
Dummy variables 𝑥 𝑤 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 noel 3952 . . . 4 ¬ 𝐹 ∈ ∅
2 pmtrrn.r . . . . . 6 𝑅 = ran 𝑇
3 pmtrrn.t . . . . . . . . 9 𝑇 = (pmTrsp‘𝐷)
4 fvprc 6223 . . . . . . . . 9 𝐷 ∈ V → (pmTrsp‘𝐷) = ∅)
53, 4syl5eq 2697 . . . . . . . 8 𝐷 ∈ V → 𝑇 = ∅)
65rneqd 5385 . . . . . . 7 𝐷 ∈ V → ran 𝑇 = ran ∅)
7 rn0 5409 . . . . . . 7 ran ∅ = ∅
86, 7syl6eq 2701 . . . . . 6 𝐷 ∈ V → ran 𝑇 = ∅)
92, 8syl5eq 2697 . . . . 5 𝐷 ∈ V → 𝑅 = ∅)
109eleq2d 2716 . . . 4 𝐷 ∈ V → (𝐹𝑅𝐹 ∈ ∅))
111, 10mtbiri 316 . . 3 𝐷 ∈ V → ¬ 𝐹𝑅)
1211con4i 113 . 2 (𝐹𝑅𝐷 ∈ V)
13 mptexg 6525 . . . . . . . 8 (𝐷 ∈ V → (𝑧𝐷 ↦ if(𝑧𝑤, (𝑤 ∖ {𝑧}), 𝑧)) ∈ V)
1413ralrimivw 2996 . . . . . . 7 (𝐷 ∈ V → ∀𝑤 ∈ {𝑥 ∈ 𝒫 𝐷𝑥 ≈ 2𝑜} (𝑧𝐷 ↦ if(𝑧𝑤, (𝑤 ∖ {𝑧}), 𝑧)) ∈ V)
15 eqid 2651 . . . . . . . 8 (𝑤 ∈ {𝑥 ∈ 𝒫 𝐷𝑥 ≈ 2𝑜} ↦ (𝑧𝐷 ↦ if(𝑧𝑤, (𝑤 ∖ {𝑧}), 𝑧))) = (𝑤 ∈ {𝑥 ∈ 𝒫 𝐷𝑥 ≈ 2𝑜} ↦ (𝑧𝐷 ↦ if(𝑧𝑤, (𝑤 ∖ {𝑧}), 𝑧)))
1615fnmpt 6058 . . . . . . 7 (∀𝑤 ∈ {𝑥 ∈ 𝒫 𝐷𝑥 ≈ 2𝑜} (𝑧𝐷 ↦ if(𝑧𝑤, (𝑤 ∖ {𝑧}), 𝑧)) ∈ V → (𝑤 ∈ {𝑥 ∈ 𝒫 𝐷𝑥 ≈ 2𝑜} ↦ (𝑧𝐷 ↦ if(𝑧𝑤, (𝑤 ∖ {𝑧}), 𝑧))) Fn {𝑥 ∈ 𝒫 𝐷𝑥 ≈ 2𝑜})
1714, 16syl 17 . . . . . 6 (𝐷 ∈ V → (𝑤 ∈ {𝑥 ∈ 𝒫 𝐷𝑥 ≈ 2𝑜} ↦ (𝑧𝐷 ↦ if(𝑧𝑤, (𝑤 ∖ {𝑧}), 𝑧))) Fn {𝑥 ∈ 𝒫 𝐷𝑥 ≈ 2𝑜})
183pmtrfval 17916 . . . . . . 7 (𝐷 ∈ V → 𝑇 = (𝑤 ∈ {𝑥 ∈ 𝒫 𝐷𝑥 ≈ 2𝑜} ↦ (𝑧𝐷 ↦ if(𝑧𝑤, (𝑤 ∖ {𝑧}), 𝑧))))
1918fneq1d 6019 . . . . . 6 (𝐷 ∈ V → (𝑇 Fn {𝑥 ∈ 𝒫 𝐷𝑥 ≈ 2𝑜} ↔ (𝑤 ∈ {𝑥 ∈ 𝒫 𝐷𝑥 ≈ 2𝑜} ↦ (𝑧𝐷 ↦ if(𝑧𝑤, (𝑤 ∖ {𝑧}), 𝑧))) Fn {𝑥 ∈ 𝒫 𝐷𝑥 ≈ 2𝑜}))
2017, 19mpbird 247 . . . . 5 (𝐷 ∈ V → 𝑇 Fn {𝑥 ∈ 𝒫 𝐷𝑥 ≈ 2𝑜})
21 fvelrnb 6282 . . . . 5 (𝑇 Fn {𝑥 ∈ 𝒫 𝐷𝑥 ≈ 2𝑜} → (𝐹 ∈ ran 𝑇 ↔ ∃𝑦 ∈ {𝑥 ∈ 𝒫 𝐷𝑥 ≈ 2𝑜} (𝑇𝑦) = 𝐹))
2220, 21syl 17 . . . 4 (𝐷 ∈ V → (𝐹 ∈ ran 𝑇 ↔ ∃𝑦 ∈ {𝑥 ∈ 𝒫 𝐷𝑥 ≈ 2𝑜} (𝑇𝑦) = 𝐹))
232eleq2i 2722 . . . 4 (𝐹𝑅𝐹 ∈ ran 𝑇)
24 breq1 4688 . . . . . 6 (𝑥 = 𝑦 → (𝑥 ≈ 2𝑜𝑦 ≈ 2𝑜))
2524rexrab 3403 . . . . 5 (∃𝑦 ∈ {𝑥 ∈ 𝒫 𝐷𝑥 ≈ 2𝑜} (𝑇𝑦) = 𝐹 ↔ ∃𝑦 ∈ 𝒫 𝐷(𝑦 ≈ 2𝑜 ∧ (𝑇𝑦) = 𝐹))
2625bicomi 214 . . . 4 (∃𝑦 ∈ 𝒫 𝐷(𝑦 ≈ 2𝑜 ∧ (𝑇𝑦) = 𝐹) ↔ ∃𝑦 ∈ {𝑥 ∈ 𝒫 𝐷𝑥 ≈ 2𝑜} (𝑇𝑦) = 𝐹)
2722, 23, 263bitr4g 303 . . 3 (𝐷 ∈ V → (𝐹𝑅 ↔ ∃𝑦 ∈ 𝒫 𝐷(𝑦 ≈ 2𝑜 ∧ (𝑇𝑦) = 𝐹)))
28 elpwi 4201 . . . . 5 (𝑦 ∈ 𝒫 𝐷𝑦𝐷)
29 simp1 1081 . . . . . . . . . 10 ((𝐷 ∈ V ∧ 𝑦𝐷𝑦 ≈ 2𝑜) → 𝐷 ∈ V)
303pmtrmvd 17922 . . . . . . . . . . 11 ((𝐷 ∈ V ∧ 𝑦𝐷𝑦 ≈ 2𝑜) → dom ((𝑇𝑦) ∖ I ) = 𝑦)
31 simp2 1082 . . . . . . . . . . 11 ((𝐷 ∈ V ∧ 𝑦𝐷𝑦 ≈ 2𝑜) → 𝑦𝐷)
3230, 31eqsstrd 3672 . . . . . . . . . 10 ((𝐷 ∈ V ∧ 𝑦𝐷𝑦 ≈ 2𝑜) → dom ((𝑇𝑦) ∖ I ) ⊆ 𝐷)
33 simp3 1083 . . . . . . . . . . 11 ((𝐷 ∈ V ∧ 𝑦𝐷𝑦 ≈ 2𝑜) → 𝑦 ≈ 2𝑜)
3430, 33eqbrtrd 4707 . . . . . . . . . 10 ((𝐷 ∈ V ∧ 𝑦𝐷𝑦 ≈ 2𝑜) → dom ((𝑇𝑦) ∖ I ) ≈ 2𝑜)
3529, 32, 343jca 1261 . . . . . . . . 9 ((𝐷 ∈ V ∧ 𝑦𝐷𝑦 ≈ 2𝑜) → (𝐷 ∈ V ∧ dom ((𝑇𝑦) ∖ I ) ⊆ 𝐷 ∧ dom ((𝑇𝑦) ∖ I ) ≈ 2𝑜))
3630eqcomd 2657 . . . . . . . . . 10 ((𝐷 ∈ V ∧ 𝑦𝐷𝑦 ≈ 2𝑜) → 𝑦 = dom ((𝑇𝑦) ∖ I ))
3736fveq2d 6233 . . . . . . . . 9 ((𝐷 ∈ V ∧ 𝑦𝐷𝑦 ≈ 2𝑜) → (𝑇𝑦) = (𝑇‘dom ((𝑇𝑦) ∖ I )))
3835, 37jca 553 . . . . . . . 8 ((𝐷 ∈ V ∧ 𝑦𝐷𝑦 ≈ 2𝑜) → ((𝐷 ∈ V ∧ dom ((𝑇𝑦) ∖ I ) ⊆ 𝐷 ∧ dom ((𝑇𝑦) ∖ I ) ≈ 2𝑜) ∧ (𝑇𝑦) = (𝑇‘dom ((𝑇𝑦) ∖ I ))))
39 difeq1 3754 . . . . . . . . . . 11 ((𝑇𝑦) = 𝐹 → ((𝑇𝑦) ∖ I ) = (𝐹 ∖ I ))
4039dmeqd 5358 . . . . . . . . . 10 ((𝑇𝑦) = 𝐹 → dom ((𝑇𝑦) ∖ I ) = dom (𝐹 ∖ I ))
41 pmtrfrn.p . . . . . . . . . 10 𝑃 = dom (𝐹 ∖ I )
4240, 41syl6eqr 2703 . . . . . . . . 9 ((𝑇𝑦) = 𝐹 → dom ((𝑇𝑦) ∖ I ) = 𝑃)
43 sseq1 3659 . . . . . . . . . . . 12 (dom ((𝑇𝑦) ∖ I ) = 𝑃 → (dom ((𝑇𝑦) ∖ I ) ⊆ 𝐷𝑃𝐷))
44 breq1 4688 . . . . . . . . . . . 12 (dom ((𝑇𝑦) ∖ I ) = 𝑃 → (dom ((𝑇𝑦) ∖ I ) ≈ 2𝑜𝑃 ≈ 2𝑜))
4543, 443anbi23d 1442 . . . . . . . . . . 11 (dom ((𝑇𝑦) ∖ I ) = 𝑃 → ((𝐷 ∈ V ∧ dom ((𝑇𝑦) ∖ I ) ⊆ 𝐷 ∧ dom ((𝑇𝑦) ∖ I ) ≈ 2𝑜) ↔ (𝐷 ∈ V ∧ 𝑃𝐷𝑃 ≈ 2𝑜)))
4645adantl 481 . . . . . . . . . 10 (((𝑇𝑦) = 𝐹 ∧ dom ((𝑇𝑦) ∖ I ) = 𝑃) → ((𝐷 ∈ V ∧ dom ((𝑇𝑦) ∖ I ) ⊆ 𝐷 ∧ dom ((𝑇𝑦) ∖ I ) ≈ 2𝑜) ↔ (𝐷 ∈ V ∧ 𝑃𝐷𝑃 ≈ 2𝑜)))
47 simpl 472 . . . . . . . . . . 11 (((𝑇𝑦) = 𝐹 ∧ dom ((𝑇𝑦) ∖ I ) = 𝑃) → (𝑇𝑦) = 𝐹)
48 fveq2 6229 . . . . . . . . . . . 12 (dom ((𝑇𝑦) ∖ I ) = 𝑃 → (𝑇‘dom ((𝑇𝑦) ∖ I )) = (𝑇𝑃))
4948adantl 481 . . . . . . . . . . 11 (((𝑇𝑦) = 𝐹 ∧ dom ((𝑇𝑦) ∖ I ) = 𝑃) → (𝑇‘dom ((𝑇𝑦) ∖ I )) = (𝑇𝑃))
5047, 49eqeq12d 2666 . . . . . . . . . 10 (((𝑇𝑦) = 𝐹 ∧ dom ((𝑇𝑦) ∖ I ) = 𝑃) → ((𝑇𝑦) = (𝑇‘dom ((𝑇𝑦) ∖ I )) ↔ 𝐹 = (𝑇𝑃)))
5146, 50anbi12d 747 . . . . . . . . 9 (((𝑇𝑦) = 𝐹 ∧ dom ((𝑇𝑦) ∖ I ) = 𝑃) → (((𝐷 ∈ V ∧ dom ((𝑇𝑦) ∖ I ) ⊆ 𝐷 ∧ dom ((𝑇𝑦) ∖ I ) ≈ 2𝑜) ∧ (𝑇𝑦) = (𝑇‘dom ((𝑇𝑦) ∖ I ))) ↔ ((𝐷 ∈ V ∧ 𝑃𝐷𝑃 ≈ 2𝑜) ∧ 𝐹 = (𝑇𝑃))))
5242, 51mpdan 703 . . . . . . . 8 ((𝑇𝑦) = 𝐹 → (((𝐷 ∈ V ∧ dom ((𝑇𝑦) ∖ I ) ⊆ 𝐷 ∧ dom ((𝑇𝑦) ∖ I ) ≈ 2𝑜) ∧ (𝑇𝑦) = (𝑇‘dom ((𝑇𝑦) ∖ I ))) ↔ ((𝐷 ∈ V ∧ 𝑃𝐷𝑃 ≈ 2𝑜) ∧ 𝐹 = (𝑇𝑃))))
5338, 52syl5ibcom 235 . . . . . . 7 ((𝐷 ∈ V ∧ 𝑦𝐷𝑦 ≈ 2𝑜) → ((𝑇𝑦) = 𝐹 → ((𝐷 ∈ V ∧ 𝑃𝐷𝑃 ≈ 2𝑜) ∧ 𝐹 = (𝑇𝑃))))
54533exp 1283 . . . . . 6 (𝐷 ∈ V → (𝑦𝐷 → (𝑦 ≈ 2𝑜 → ((𝑇𝑦) = 𝐹 → ((𝐷 ∈ V ∧ 𝑃𝐷𝑃 ≈ 2𝑜) ∧ 𝐹 = (𝑇𝑃))))))
5554imp4a 613 . . . . 5 (𝐷 ∈ V → (𝑦𝐷 → ((𝑦 ≈ 2𝑜 ∧ (𝑇𝑦) = 𝐹) → ((𝐷 ∈ V ∧ 𝑃𝐷𝑃 ≈ 2𝑜) ∧ 𝐹 = (𝑇𝑃)))))
5628, 55syl5 34 . . . 4 (𝐷 ∈ V → (𝑦 ∈ 𝒫 𝐷 → ((𝑦 ≈ 2𝑜 ∧ (𝑇𝑦) = 𝐹) → ((𝐷 ∈ V ∧ 𝑃𝐷𝑃 ≈ 2𝑜) ∧ 𝐹 = (𝑇𝑃)))))
5756rexlimdv 3059 . . 3 (𝐷 ∈ V → (∃𝑦 ∈ 𝒫 𝐷(𝑦 ≈ 2𝑜 ∧ (𝑇𝑦) = 𝐹) → ((𝐷 ∈ V ∧ 𝑃𝐷𝑃 ≈ 2𝑜) ∧ 𝐹 = (𝑇𝑃))))
5827, 57sylbid 230 . 2 (𝐷 ∈ V → (𝐹𝑅 → ((𝐷 ∈ V ∧ 𝑃𝐷𝑃 ≈ 2𝑜) ∧ 𝐹 = (𝑇𝑃))))
5912, 58mpcom 38 1 (𝐹𝑅 → ((𝐷 ∈ V ∧ 𝑃𝐷𝑃 ≈ 2𝑜) ∧ 𝐹 = (𝑇𝑃)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 383  w3a 1054   = wceq 1523  wcel 2030  wral 2941  wrex 2942  {crab 2945  Vcvv 3231  cdif 3604  wss 3607  c0 3948  ifcif 4119  𝒫 cpw 4191  {csn 4210   cuni 4468   class class class wbr 4685  cmpt 4762   I cid 5052  dom cdm 5143  ran crn 5144   Fn wfn 5921  cfv 5926  2𝑜c2o 7599  cen 7994  pmTrspcpmtr 17907
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-om 7108  df-1o 7605  df-2o 7606  df-er 7787  df-en 7998  df-fin 8001  df-pmtr 17908
This theorem is referenced by:  pmtrffv  17925  pmtrrn2  17926  pmtrfinv  17927  pmtrfmvdn0  17928  pmtrff1o  17929  pmtrfcnv  17930  pmtrfb  17931
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