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Theorem pmtrfrn 18578
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 4247 . . . 4 ¬ 𝐹 ∈ ∅
2 pmtrrn.r . . . . . 6 𝑅 = ran 𝑇
3 pmtrrn.t . . . . . . 7 𝑇 = (pmTrsp‘𝐷)
43rnfvprc 6639 . . . . . 6 𝐷 ∈ V → ran 𝑇 = ∅)
52, 4syl5eq 2845 . . . . 5 𝐷 ∈ V → 𝑅 = ∅)
65eleq2d 2875 . . . 4 𝐷 ∈ V → (𝐹𝑅𝐹 ∈ ∅))
71, 6mtbiri 330 . . 3 𝐷 ∈ V → ¬ 𝐹𝑅)
87con4i 114 . 2 (𝐹𝑅𝐷 ∈ V)
9 mptexg 6961 . . . . . . . 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 2798 . . . . . . . 8 (𝑤 ∈ {𝑥 ∈ 𝒫 𝐷𝑥 ≈ 2o} ↦ (𝑧𝐷 ↦ if(𝑧𝑤, (𝑤 ∖ {𝑧}), 𝑧))) = (𝑤 ∈ {𝑥 ∈ 𝒫 𝐷𝑥 ≈ 2o} ↦ (𝑧𝐷 ↦ if(𝑧𝑤, (𝑤 ∖ {𝑧}), 𝑧)))
1211fnmpt 6460 . . . . . . 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 18570 . . . . . . 7 (𝐷 ∈ V → 𝑇 = (𝑤 ∈ {𝑥 ∈ 𝒫 𝐷𝑥 ≈ 2o} ↦ (𝑧𝐷 ↦ if(𝑧𝑤, (𝑤 ∖ {𝑧}), 𝑧))))
1514fneq1d 6416 . . . . . 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 260 . . . . 5 (𝐷 ∈ V → 𝑇 Fn {𝑥 ∈ 𝒫 𝐷𝑥 ≈ 2o})
17 fvelrnb 6701 . . . . 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 2881 . . . 4 (𝐹𝑅𝐹 ∈ ran 𝑇)
20 breq1 5033 . . . . . 6 (𝑥 = 𝑦 → (𝑥 ≈ 2o𝑦 ≈ 2o))
2120rexrab 3635 . . . . 5 (∃𝑦 ∈ {𝑥 ∈ 𝒫 𝐷𝑥 ≈ 2o} (𝑇𝑦) = 𝐹 ↔ ∃𝑦 ∈ 𝒫 𝐷(𝑦 ≈ 2o ∧ (𝑇𝑦) = 𝐹))
2221bicomi 227 . . . 4 (∃𝑦 ∈ 𝒫 𝐷(𝑦 ≈ 2o ∧ (𝑇𝑦) = 𝐹) ↔ ∃𝑦 ∈ {𝑥 ∈ 𝒫 𝐷𝑥 ≈ 2o} (𝑇𝑦) = 𝐹)
2318, 19, 223bitr4g 317 . . 3 (𝐷 ∈ V → (𝐹𝑅 ↔ ∃𝑦 ∈ 𝒫 𝐷(𝑦 ≈ 2o ∧ (𝑇𝑦) = 𝐹)))
24 elpwi 4506 . . . . 5 (𝑦 ∈ 𝒫 𝐷𝑦𝐷)
25 simp1 1133 . . . . . . . . . 10 ((𝐷 ∈ V ∧ 𝑦𝐷𝑦 ≈ 2o) → 𝐷 ∈ V)
263pmtrmvd 18576 . . . . . . . . . . 11 ((𝐷 ∈ V ∧ 𝑦𝐷𝑦 ≈ 2o) → dom ((𝑇𝑦) ∖ I ) = 𝑦)
27 simp2 1134 . . . . . . . . . . 11 ((𝐷 ∈ V ∧ 𝑦𝐷𝑦 ≈ 2o) → 𝑦𝐷)
2826, 27eqsstrd 3953 . . . . . . . . . 10 ((𝐷 ∈ V ∧ 𝑦𝐷𝑦 ≈ 2o) → dom ((𝑇𝑦) ∖ I ) ⊆ 𝐷)
29 simp3 1135 . . . . . . . . . . 11 ((𝐷 ∈ V ∧ 𝑦𝐷𝑦 ≈ 2o) → 𝑦 ≈ 2o)
3026, 29eqbrtrd 5052 . . . . . . . . . 10 ((𝐷 ∈ V ∧ 𝑦𝐷𝑦 ≈ 2o) → dom ((𝑇𝑦) ∖ I ) ≈ 2o)
3125, 28, 303jca 1125 . . . . . . . . 9 ((𝐷 ∈ V ∧ 𝑦𝐷𝑦 ≈ 2o) → (𝐷 ∈ V ∧ dom ((𝑇𝑦) ∖ I ) ⊆ 𝐷 ∧ dom ((𝑇𝑦) ∖ I ) ≈ 2o))
3226eqcomd 2804 . . . . . . . . . 10 ((𝐷 ∈ V ∧ 𝑦𝐷𝑦 ≈ 2o) → 𝑦 = dom ((𝑇𝑦) ∖ I ))
3332fveq2d 6649 . . . . . . . . 9 ((𝐷 ∈ V ∧ 𝑦𝐷𝑦 ≈ 2o) → (𝑇𝑦) = (𝑇‘dom ((𝑇𝑦) ∖ I )))
3431, 33jca 515 . . . . . . . 8 ((𝐷 ∈ V ∧ 𝑦𝐷𝑦 ≈ 2o) → ((𝐷 ∈ V ∧ dom ((𝑇𝑦) ∖ I ) ⊆ 𝐷 ∧ dom ((𝑇𝑦) ∖ I ) ≈ 2o) ∧ (𝑇𝑦) = (𝑇‘dom ((𝑇𝑦) ∖ I ))))
35 difeq1 4043 . . . . . . . . . . 11 ((𝑇𝑦) = 𝐹 → ((𝑇𝑦) ∖ I ) = (𝐹 ∖ I ))
3635dmeqd 5738 . . . . . . . . . 10 ((𝑇𝑦) = 𝐹 → dom ((𝑇𝑦) ∖ I ) = dom (𝐹 ∖ I ))
37 pmtrfrn.p . . . . . . . . . 10 𝑃 = dom (𝐹 ∖ I )
3836, 37eqtr4di 2851 . . . . . . . . 9 ((𝑇𝑦) = 𝐹 → dom ((𝑇𝑦) ∖ I ) = 𝑃)
39 sseq1 3940 . . . . . . . . . . . 12 (dom ((𝑇𝑦) ∖ I ) = 𝑃 → (dom ((𝑇𝑦) ∖ I ) ⊆ 𝐷𝑃𝐷))
40 breq1 5033 . . . . . . . . . . . 12 (dom ((𝑇𝑦) ∖ I ) = 𝑃 → (dom ((𝑇𝑦) ∖ I ) ≈ 2o𝑃 ≈ 2o))
4139, 403anbi23d 1436 . . . . . . . . . . 11 (dom ((𝑇𝑦) ∖ I ) = 𝑃 → ((𝐷 ∈ V ∧ dom ((𝑇𝑦) ∖ I ) ⊆ 𝐷 ∧ dom ((𝑇𝑦) ∖ I ) ≈ 2o) ↔ (𝐷 ∈ V ∧ 𝑃𝐷𝑃 ≈ 2o)))
4241adantl 485 . . . . . . . . . 10 (((𝑇𝑦) = 𝐹 ∧ dom ((𝑇𝑦) ∖ I ) = 𝑃) → ((𝐷 ∈ V ∧ dom ((𝑇𝑦) ∖ I ) ⊆ 𝐷 ∧ dom ((𝑇𝑦) ∖ I ) ≈ 2o) ↔ (𝐷 ∈ V ∧ 𝑃𝐷𝑃 ≈ 2o)))
43 simpl 486 . . . . . . . . . . 11 (((𝑇𝑦) = 𝐹 ∧ dom ((𝑇𝑦) ∖ I ) = 𝑃) → (𝑇𝑦) = 𝐹)
44 fveq2 6645 . . . . . . . . . . . 12 (dom ((𝑇𝑦) ∖ I ) = 𝑃 → (𝑇‘dom ((𝑇𝑦) ∖ I )) = (𝑇𝑃))
4544adantl 485 . . . . . . . . . . 11 (((𝑇𝑦) = 𝐹 ∧ dom ((𝑇𝑦) ∖ I ) = 𝑃) → (𝑇‘dom ((𝑇𝑦) ∖ I )) = (𝑇𝑃))
4643, 45eqeq12d 2814 . . . . . . . . . 10 (((𝑇𝑦) = 𝐹 ∧ dom ((𝑇𝑦) ∖ I ) = 𝑃) → ((𝑇𝑦) = (𝑇‘dom ((𝑇𝑦) ∖ I )) ↔ 𝐹 = (𝑇𝑃)))
4742, 46anbi12d 633 . . . . . . . . 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 686 . . . . . . . 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 248 . . . . . . 7 ((𝐷 ∈ V ∧ 𝑦𝐷𝑦 ≈ 2o) → ((𝑇𝑦) = 𝐹 → ((𝐷 ∈ V ∧ 𝑃𝐷𝑃 ≈ 2o) ∧ 𝐹 = (𝑇𝑃))))
50493exp 1116 . . . . . 6 (𝐷 ∈ V → (𝑦𝐷 → (𝑦 ≈ 2o → ((𝑇𝑦) = 𝐹 → ((𝐷 ∈ V ∧ 𝑃𝐷𝑃 ≈ 2o) ∧ 𝐹 = (𝑇𝑃))))))
5150imp4a 426 . . . . 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 3242 . . 3 (𝐷 ∈ V → (∃𝑦 ∈ 𝒫 𝐷(𝑦 ≈ 2o ∧ (𝑇𝑦) = 𝐹) → ((𝐷 ∈ V ∧ 𝑃𝐷𝑃 ≈ 2o) ∧ 𝐹 = (𝑇𝑃))))
5423, 53sylbid 243 . 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 209  wa 399  w3a 1084   = wceq 1538  wcel 2111  wral 3106  wrex 3107  {crab 3110  Vcvv 3441  cdif 3878  wss 3881  c0 4243  ifcif 4425  𝒫 cpw 4497  {csn 4525   cuni 4800   class class class wbr 5030  cmpt 5110   I cid 5424  dom cdm 5519  ran crn 5520   Fn wfn 6319  cfv 6324  2oc2o 8079  cen 8489  pmTrspcpmtr 18561
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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-reu 3113  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-pss 3900  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4801  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-tr 5137  df-id 5425  df-eprel 5430  df-po 5438  df-so 5439  df-fr 5478  df-we 5480  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-ord 6162  df-on 6163  df-lim 6164  df-suc 6165  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332  df-om 7561  df-1o 8085  df-2o 8086  df-er 8272  df-en 8493  df-fin 8496  df-pmtr 18562
This theorem is referenced by:  pmtrffv  18579  pmtrrn2  18580  pmtrfinv  18581  pmtrfmvdn0  18582  pmtrff1o  18583  pmtrfcnv  18584  pmtrfb  18585
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