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Theorem pmtrfrn 19320
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 4329 . . . 4 ¬ 𝐹 ∈ ∅
2 pmtrrn.r . . . . . 6 𝑅 = ran 𝑇
3 pmtrrn.t . . . . . . 7 𝑇 = (pmTrsp‘𝐷)
43rnfvprc 6882 . . . . . 6 𝐷 ∈ V → ran 𝑇 = ∅)
52, 4eqtrid 2784 . . . . 5 𝐷 ∈ V → 𝑅 = ∅)
65eleq2d 2819 . . . 4 𝐷 ∈ V → (𝐹𝑅𝐹 ∈ ∅))
71, 6mtbiri 326 . . 3 𝐷 ∈ V → ¬ 𝐹𝑅)
87con4i 114 . 2 (𝐹𝑅𝐷 ∈ V)
9 mptexg 7219 . . . . . . . 8 (𝐷 ∈ V → (𝑧𝐷 ↦ if(𝑧𝑤, (𝑤 ∖ {𝑧}), 𝑧)) ∈ V)
109ralrimivw 3150 . . . . . . 7 (𝐷 ∈ V → ∀𝑤 ∈ {𝑥 ∈ 𝒫 𝐷𝑥 ≈ 2o} (𝑧𝐷 ↦ if(𝑧𝑤, (𝑤 ∖ {𝑧}), 𝑧)) ∈ V)
11 eqid 2732 . . . . . . . 8 (𝑤 ∈ {𝑥 ∈ 𝒫 𝐷𝑥 ≈ 2o} ↦ (𝑧𝐷 ↦ if(𝑧𝑤, (𝑤 ∖ {𝑧}), 𝑧))) = (𝑤 ∈ {𝑥 ∈ 𝒫 𝐷𝑥 ≈ 2o} ↦ (𝑧𝐷 ↦ if(𝑧𝑤, (𝑤 ∖ {𝑧}), 𝑧)))
1211fnmpt 6687 . . . . . . 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})
1310, 12syl 17 . . . . . 6 (𝐷 ∈ V → (𝑤 ∈ {𝑥 ∈ 𝒫 𝐷𝑥 ≈ 2o} ↦ (𝑧𝐷 ↦ if(𝑧𝑤, (𝑤 ∖ {𝑧}), 𝑧))) Fn {𝑥 ∈ 𝒫 𝐷𝑥 ≈ 2o})
143pmtrfval 19312 . . . . . . 7 (𝐷 ∈ V → 𝑇 = (𝑤 ∈ {𝑥 ∈ 𝒫 𝐷𝑥 ≈ 2o} ↦ (𝑧𝐷 ↦ if(𝑧𝑤, (𝑤 ∖ {𝑧}), 𝑧))))
1514fneq1d 6639 . . . . . 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}))
1613, 15mpbird 256 . . . . 5 (𝐷 ∈ V → 𝑇 Fn {𝑥 ∈ 𝒫 𝐷𝑥 ≈ 2o})
17 fvelrnb 6949 . . . . 5 (𝑇 Fn {𝑥 ∈ 𝒫 𝐷𝑥 ≈ 2o} → (𝐹 ∈ ran 𝑇 ↔ ∃𝑦 ∈ {𝑥 ∈ 𝒫 𝐷𝑥 ≈ 2o} (𝑇𝑦) = 𝐹))
1816, 17syl 17 . . . 4 (𝐷 ∈ V → (𝐹 ∈ ran 𝑇 ↔ ∃𝑦 ∈ {𝑥 ∈ 𝒫 𝐷𝑥 ≈ 2o} (𝑇𝑦) = 𝐹))
192eleq2i 2825 . . . 4 (𝐹𝑅𝐹 ∈ ran 𝑇)
20 breq1 5150 . . . . . 6 (𝑥 = 𝑦 → (𝑥 ≈ 2o𝑦 ≈ 2o))
2120rexrab 3691 . . . . 5 (∃𝑦 ∈ {𝑥 ∈ 𝒫 𝐷𝑥 ≈ 2o} (𝑇𝑦) = 𝐹 ↔ ∃𝑦 ∈ 𝒫 𝐷(𝑦 ≈ 2o ∧ (𝑇𝑦) = 𝐹))
2221bicomi 223 . . . 4 (∃𝑦 ∈ 𝒫 𝐷(𝑦 ≈ 2o ∧ (𝑇𝑦) = 𝐹) ↔ ∃𝑦 ∈ {𝑥 ∈ 𝒫 𝐷𝑥 ≈ 2o} (𝑇𝑦) = 𝐹)
2318, 19, 223bitr4g 313 . . 3 (𝐷 ∈ V → (𝐹𝑅 ↔ ∃𝑦 ∈ 𝒫 𝐷(𝑦 ≈ 2o ∧ (𝑇𝑦) = 𝐹)))
24 elpwi 4608 . . . . 5 (𝑦 ∈ 𝒫 𝐷𝑦𝐷)
25 simp1 1136 . . . . . . . . . 10 ((𝐷 ∈ V ∧ 𝑦𝐷𝑦 ≈ 2o) → 𝐷 ∈ V)
263pmtrmvd 19318 . . . . . . . . . . 11 ((𝐷 ∈ V ∧ 𝑦𝐷𝑦 ≈ 2o) → dom ((𝑇𝑦) ∖ I ) = 𝑦)
27 simp2 1137 . . . . . . . . . . 11 ((𝐷 ∈ V ∧ 𝑦𝐷𝑦 ≈ 2o) → 𝑦𝐷)
2826, 27eqsstrd 4019 . . . . . . . . . 10 ((𝐷 ∈ V ∧ 𝑦𝐷𝑦 ≈ 2o) → dom ((𝑇𝑦) ∖ I ) ⊆ 𝐷)
29 simp3 1138 . . . . . . . . . . 11 ((𝐷 ∈ V ∧ 𝑦𝐷𝑦 ≈ 2o) → 𝑦 ≈ 2o)
3026, 29eqbrtrd 5169 . . . . . . . . . 10 ((𝐷 ∈ V ∧ 𝑦𝐷𝑦 ≈ 2o) → dom ((𝑇𝑦) ∖ I ) ≈ 2o)
3125, 28, 303jca 1128 . . . . . . . . 9 ((𝐷 ∈ V ∧ 𝑦𝐷𝑦 ≈ 2o) → (𝐷 ∈ V ∧ dom ((𝑇𝑦) ∖ I ) ⊆ 𝐷 ∧ dom ((𝑇𝑦) ∖ I ) ≈ 2o))
3226eqcomd 2738 . . . . . . . . . 10 ((𝐷 ∈ V ∧ 𝑦𝐷𝑦 ≈ 2o) → 𝑦 = dom ((𝑇𝑦) ∖ I ))
3332fveq2d 6892 . . . . . . . . 9 ((𝐷 ∈ V ∧ 𝑦𝐷𝑦 ≈ 2o) → (𝑇𝑦) = (𝑇‘dom ((𝑇𝑦) ∖ I )))
3431, 33jca 512 . . . . . . . 8 ((𝐷 ∈ V ∧ 𝑦𝐷𝑦 ≈ 2o) → ((𝐷 ∈ V ∧ dom ((𝑇𝑦) ∖ I ) ⊆ 𝐷 ∧ dom ((𝑇𝑦) ∖ I ) ≈ 2o) ∧ (𝑇𝑦) = (𝑇‘dom ((𝑇𝑦) ∖ I ))))
35 difeq1 4114 . . . . . . . . . . 11 ((𝑇𝑦) = 𝐹 → ((𝑇𝑦) ∖ I ) = (𝐹 ∖ I ))
3635dmeqd 5903 . . . . . . . . . 10 ((𝑇𝑦) = 𝐹 → dom ((𝑇𝑦) ∖ I ) = dom (𝐹 ∖ I ))
37 pmtrfrn.p . . . . . . . . . 10 𝑃 = dom (𝐹 ∖ I )
3836, 37eqtr4di 2790 . . . . . . . . 9 ((𝑇𝑦) = 𝐹 → dom ((𝑇𝑦) ∖ I ) = 𝑃)
39 sseq1 4006 . . . . . . . . . . . 12 (dom ((𝑇𝑦) ∖ I ) = 𝑃 → (dom ((𝑇𝑦) ∖ I ) ⊆ 𝐷𝑃𝐷))
40 breq1 5150 . . . . . . . . . . . 12 (dom ((𝑇𝑦) ∖ I ) = 𝑃 → (dom ((𝑇𝑦) ∖ I ) ≈ 2o𝑃 ≈ 2o))
4139, 403anbi23d 1439 . . . . . . . . . . 11 (dom ((𝑇𝑦) ∖ I ) = 𝑃 → ((𝐷 ∈ V ∧ dom ((𝑇𝑦) ∖ I ) ⊆ 𝐷 ∧ dom ((𝑇𝑦) ∖ I ) ≈ 2o) ↔ (𝐷 ∈ V ∧ 𝑃𝐷𝑃 ≈ 2o)))
4241adantl 482 . . . . . . . . . 10 (((𝑇𝑦) = 𝐹 ∧ dom ((𝑇𝑦) ∖ I ) = 𝑃) → ((𝐷 ∈ V ∧ dom ((𝑇𝑦) ∖ I ) ⊆ 𝐷 ∧ dom ((𝑇𝑦) ∖ I ) ≈ 2o) ↔ (𝐷 ∈ V ∧ 𝑃𝐷𝑃 ≈ 2o)))
43 simpl 483 . . . . . . . . . . 11 (((𝑇𝑦) = 𝐹 ∧ dom ((𝑇𝑦) ∖ I ) = 𝑃) → (𝑇𝑦) = 𝐹)
44 fveq2 6888 . . . . . . . . . . . 12 (dom ((𝑇𝑦) ∖ I ) = 𝑃 → (𝑇‘dom ((𝑇𝑦) ∖ I )) = (𝑇𝑃))
4544adantl 482 . . . . . . . . . . 11 (((𝑇𝑦) = 𝐹 ∧ dom ((𝑇𝑦) ∖ I ) = 𝑃) → (𝑇‘dom ((𝑇𝑦) ∖ I )) = (𝑇𝑃))
4643, 45eqeq12d 2748 . . . . . . . . . 10 (((𝑇𝑦) = 𝐹 ∧ dom ((𝑇𝑦) ∖ I ) = 𝑃) → ((𝑇𝑦) = (𝑇‘dom ((𝑇𝑦) ∖ I )) ↔ 𝐹 = (𝑇𝑃)))
4742, 46anbi12d 631 . . . . . . . . 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))))
4838, 47mpdan 685 . . . . . . . 8 ((𝑇𝑦) = 𝐹 → (((𝐷 ∈ V ∧ dom ((𝑇𝑦) ∖ I ) ⊆ 𝐷 ∧ dom ((𝑇𝑦) ∖ I ) ≈ 2o) ∧ (𝑇𝑦) = (𝑇‘dom ((𝑇𝑦) ∖ I ))) ↔ ((𝐷 ∈ V ∧ 𝑃𝐷𝑃 ≈ 2o) ∧ 𝐹 = (𝑇𝑃))))
4934, 48syl5ibcom 244 . . . . . . 7 ((𝐷 ∈ V ∧ 𝑦𝐷𝑦 ≈ 2o) → ((𝑇𝑦) = 𝐹 → ((𝐷 ∈ V ∧ 𝑃𝐷𝑃 ≈ 2o) ∧ 𝐹 = (𝑇𝑃))))
50493exp 1119 . . . . . 6 (𝐷 ∈ V → (𝑦𝐷 → (𝑦 ≈ 2o → ((𝑇𝑦) = 𝐹 → ((𝐷 ∈ V ∧ 𝑃𝐷𝑃 ≈ 2o) ∧ 𝐹 = (𝑇𝑃))))))
5150imp4a 423 . . . . 5 (𝐷 ∈ V → (𝑦𝐷 → ((𝑦 ≈ 2o ∧ (𝑇𝑦) = 𝐹) → ((𝐷 ∈ V ∧ 𝑃𝐷𝑃 ≈ 2o) ∧ 𝐹 = (𝑇𝑃)))))
5224, 51syl5 34 . . . 4 (𝐷 ∈ V → (𝑦 ∈ 𝒫 𝐷 → ((𝑦 ≈ 2o ∧ (𝑇𝑦) = 𝐹) → ((𝐷 ∈ V ∧ 𝑃𝐷𝑃 ≈ 2o) ∧ 𝐹 = (𝑇𝑃)))))
5352rexlimdv 3153 . . 3 (𝐷 ∈ V → (∃𝑦 ∈ 𝒫 𝐷(𝑦 ≈ 2o ∧ (𝑇𝑦) = 𝐹) → ((𝐷 ∈ V ∧ 𝑃𝐷𝑃 ≈ 2o) ∧ 𝐹 = (𝑇𝑃))))
5423, 53sylbid 239 . 2 (𝐷 ∈ V → (𝐹𝑅 → ((𝐷 ∈ V ∧ 𝑃𝐷𝑃 ≈ 2o) ∧ 𝐹 = (𝑇𝑃))))
558, 54mpcom 38 1 (𝐹𝑅 → ((𝐷 ∈ V ∧ 𝑃𝐷𝑃 ≈ 2o) ∧ 𝐹 = (𝑇𝑃)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 396  w3a 1087   = wceq 1541  wcel 2106  wral 3061  wrex 3070  {crab 3432  Vcvv 3474  cdif 3944  wss 3947  c0 4321  ifcif 4527  𝒫 cpw 4601  {csn 4627   cuni 4907   class class class wbr 5147  cmpt 5230   I cid 5572  dom cdm 5675  ran crn 5676   Fn wfn 6535  cfv 6540  2oc2o 8456  cen 8932  pmTrspcpmtr 19303
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7721
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5573  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-we 5632  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-om 7852  df-1o 8462  df-2o 8463  df-en 8936  df-pmtr 19304
This theorem is referenced by:  pmtrffv  19321  pmtrrn2  19322  pmtrfinv  19323  pmtrfmvdn0  19324  pmtrff1o  19325  pmtrfcnv  19326  pmtrfb  19327
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