MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  pmtrfrn Structured version   Visualization version   GIF version

Theorem pmtrfrn 18079
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 4120 . . . 4 ¬ 𝐹 ∈ ∅
2 pmtrrn.r . . . . . 6 𝑅 = ran 𝑇
3 pmtrrn.t . . . . . . 7 𝑇 = (pmTrsp‘𝐷)
43rnfvprc 6402 . . . . . 6 𝐷 ∈ V → ran 𝑇 = ∅)
52, 4syl5eq 2852 . . . . 5 𝐷 ∈ V → 𝑅 = ∅)
65eleq2d 2871 . . . 4 𝐷 ∈ V → (𝐹𝑅𝐹 ∈ ∅))
71, 6mtbiri 318 . . 3 𝐷 ∈ V → ¬ 𝐹𝑅)
87con4i 114 . 2 (𝐹𝑅𝐷 ∈ V)
9 mptexg 6709 . . . . . . . 8 (𝐷 ∈ V → (𝑧𝐷 ↦ if(𝑧𝑤, (𝑤 ∖ {𝑧}), 𝑧)) ∈ V)
109ralrimivw 3155 . . . . . . 7 (𝐷 ∈ V → ∀𝑤 ∈ {𝑥 ∈ 𝒫 𝐷𝑥 ≈ 2𝑜} (𝑧𝐷 ↦ if(𝑧𝑤, (𝑤 ∖ {𝑧}), 𝑧)) ∈ V)
11 eqid 2806 . . . . . . . 8 (𝑤 ∈ {𝑥 ∈ 𝒫 𝐷𝑥 ≈ 2𝑜} ↦ (𝑧𝐷 ↦ if(𝑧𝑤, (𝑤 ∖ {𝑧}), 𝑧))) = (𝑤 ∈ {𝑥 ∈ 𝒫 𝐷𝑥 ≈ 2𝑜} ↦ (𝑧𝐷 ↦ if(𝑧𝑤, (𝑤 ∖ {𝑧}), 𝑧)))
1211fnmpt 6231 . . . . . . 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 18071 . . . . . . 7 (𝐷 ∈ V → 𝑇 = (𝑤 ∈ {𝑥 ∈ 𝒫 𝐷𝑥 ≈ 2𝑜} ↦ (𝑧𝐷 ↦ if(𝑧𝑤, (𝑤 ∖ {𝑧}), 𝑧))))
1514fneq1d 6192 . . . . . 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 248 . . . . 5 (𝐷 ∈ V → 𝑇 Fn {𝑥 ∈ 𝒫 𝐷𝑥 ≈ 2𝑜})
17 fvelrnb 6464 . . . . 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 2877 . . . 4 (𝐹𝑅𝐹 ∈ ran 𝑇)
20 breq1 4847 . . . . . 6 (𝑥 = 𝑦 → (𝑥 ≈ 2𝑜𝑦 ≈ 2𝑜))
2120rexrab 3566 . . . . 5 (∃𝑦 ∈ {𝑥 ∈ 𝒫 𝐷𝑥 ≈ 2𝑜} (𝑇𝑦) = 𝐹 ↔ ∃𝑦 ∈ 𝒫 𝐷(𝑦 ≈ 2𝑜 ∧ (𝑇𝑦) = 𝐹))
2221bicomi 215 . . . 4 (∃𝑦 ∈ 𝒫 𝐷(𝑦 ≈ 2𝑜 ∧ (𝑇𝑦) = 𝐹) ↔ ∃𝑦 ∈ {𝑥 ∈ 𝒫 𝐷𝑥 ≈ 2𝑜} (𝑇𝑦) = 𝐹)
2318, 19, 223bitr4g 305 . . 3 (𝐷 ∈ V → (𝐹𝑅 ↔ ∃𝑦 ∈ 𝒫 𝐷(𝑦 ≈ 2𝑜 ∧ (𝑇𝑦) = 𝐹)))
24 elpwi 4361 . . . . 5 (𝑦 ∈ 𝒫 𝐷𝑦𝐷)
25 simp1 1159 . . . . . . . . . 10 ((𝐷 ∈ V ∧ 𝑦𝐷𝑦 ≈ 2𝑜) → 𝐷 ∈ V)
263pmtrmvd 18077 . . . . . . . . . . 11 ((𝐷 ∈ V ∧ 𝑦𝐷𝑦 ≈ 2𝑜) → dom ((𝑇𝑦) ∖ I ) = 𝑦)
27 simp2 1160 . . . . . . . . . . 11 ((𝐷 ∈ V ∧ 𝑦𝐷𝑦 ≈ 2𝑜) → 𝑦𝐷)
2826, 27eqsstrd 3836 . . . . . . . . . 10 ((𝐷 ∈ V ∧ 𝑦𝐷𝑦 ≈ 2𝑜) → dom ((𝑇𝑦) ∖ I ) ⊆ 𝐷)
29 simp3 1161 . . . . . . . . . . 11 ((𝐷 ∈ V ∧ 𝑦𝐷𝑦 ≈ 2𝑜) → 𝑦 ≈ 2𝑜)
3026, 29eqbrtrd 4866 . . . . . . . . . 10 ((𝐷 ∈ V ∧ 𝑦𝐷𝑦 ≈ 2𝑜) → dom ((𝑇𝑦) ∖ I ) ≈ 2𝑜)
3125, 28, 303jca 1151 . . . . . . . . 9 ((𝐷 ∈ V ∧ 𝑦𝐷𝑦 ≈ 2𝑜) → (𝐷 ∈ V ∧ dom ((𝑇𝑦) ∖ I ) ⊆ 𝐷 ∧ dom ((𝑇𝑦) ∖ I ) ≈ 2𝑜))
3226eqcomd 2812 . . . . . . . . . 10 ((𝐷 ∈ V ∧ 𝑦𝐷𝑦 ≈ 2𝑜) → 𝑦 = dom ((𝑇𝑦) ∖ I ))
3332fveq2d 6412 . . . . . . . . 9 ((𝐷 ∈ V ∧ 𝑦𝐷𝑦 ≈ 2𝑜) → (𝑇𝑦) = (𝑇‘dom ((𝑇𝑦) ∖ I )))
3431, 33jca 503 . . . . . . . 8 ((𝐷 ∈ V ∧ 𝑦𝐷𝑦 ≈ 2𝑜) → ((𝐷 ∈ V ∧ dom ((𝑇𝑦) ∖ I ) ⊆ 𝐷 ∧ dom ((𝑇𝑦) ∖ I ) ≈ 2𝑜) ∧ (𝑇𝑦) = (𝑇‘dom ((𝑇𝑦) ∖ I ))))
35 difeq1 3920 . . . . . . . . . . 11 ((𝑇𝑦) = 𝐹 → ((𝑇𝑦) ∖ I ) = (𝐹 ∖ I ))
3635dmeqd 5527 . . . . . . . . . 10 ((𝑇𝑦) = 𝐹 → dom ((𝑇𝑦) ∖ I ) = dom (𝐹 ∖ I ))
37 pmtrfrn.p . . . . . . . . . 10 𝑃 = dom (𝐹 ∖ I )
3836, 37syl6eqr 2858 . . . . . . . . 9 ((𝑇𝑦) = 𝐹 → dom ((𝑇𝑦) ∖ I ) = 𝑃)
39 sseq1 3823 . . . . . . . . . . . 12 (dom ((𝑇𝑦) ∖ I ) = 𝑃 → (dom ((𝑇𝑦) ∖ I ) ⊆ 𝐷𝑃𝐷))
40 breq1 4847 . . . . . . . . . . . 12 (dom ((𝑇𝑦) ∖ I ) = 𝑃 → (dom ((𝑇𝑦) ∖ I ) ≈ 2𝑜𝑃 ≈ 2𝑜))
4139, 403anbi23d 1556 . . . . . . . . . . 11 (dom ((𝑇𝑦) ∖ I ) = 𝑃 → ((𝐷 ∈ V ∧ dom ((𝑇𝑦) ∖ I ) ⊆ 𝐷 ∧ dom ((𝑇𝑦) ∖ I ) ≈ 2𝑜) ↔ (𝐷 ∈ V ∧ 𝑃𝐷𝑃 ≈ 2𝑜)))
4241adantl 469 . . . . . . . . . 10 (((𝑇𝑦) = 𝐹 ∧ dom ((𝑇𝑦) ∖ I ) = 𝑃) → ((𝐷 ∈ V ∧ dom ((𝑇𝑦) ∖ I ) ⊆ 𝐷 ∧ dom ((𝑇𝑦) ∖ I ) ≈ 2𝑜) ↔ (𝐷 ∈ V ∧ 𝑃𝐷𝑃 ≈ 2𝑜)))
43 simpl 470 . . . . . . . . . . 11 (((𝑇𝑦) = 𝐹 ∧ dom ((𝑇𝑦) ∖ I ) = 𝑃) → (𝑇𝑦) = 𝐹)
44 fveq2 6408 . . . . . . . . . . . 12 (dom ((𝑇𝑦) ∖ I ) = 𝑃 → (𝑇‘dom ((𝑇𝑦) ∖ I )) = (𝑇𝑃))
4544adantl 469 . . . . . . . . . . 11 (((𝑇𝑦) = 𝐹 ∧ dom ((𝑇𝑦) ∖ I ) = 𝑃) → (𝑇‘dom ((𝑇𝑦) ∖ I )) = (𝑇𝑃))
4643, 45eqeq12d 2821 . . . . . . . . . 10 (((𝑇𝑦) = 𝐹 ∧ dom ((𝑇𝑦) ∖ I ) = 𝑃) → ((𝑇𝑦) = (𝑇‘dom ((𝑇𝑦) ∖ I )) ↔ 𝐹 = (𝑇𝑃)))
4742, 46anbi12d 618 . . . . . . . . 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 670 . . . . . . . 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 236 . . . . . . 7 ((𝐷 ∈ V ∧ 𝑦𝐷𝑦 ≈ 2𝑜) → ((𝑇𝑦) = 𝐹 → ((𝐷 ∈ V ∧ 𝑃𝐷𝑃 ≈ 2𝑜) ∧ 𝐹 = (𝑇𝑃))))
50493exp 1141 . . . . . 6 (𝐷 ∈ V → (𝑦𝐷 → (𝑦 ≈ 2𝑜 → ((𝑇𝑦) = 𝐹 → ((𝐷 ∈ V ∧ 𝑃𝐷𝑃 ≈ 2𝑜) ∧ 𝐹 = (𝑇𝑃))))))
5150imp4a 411 . . . . 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 3218 . . 3 (𝐷 ∈ V → (∃𝑦 ∈ 𝒫 𝐷(𝑦 ≈ 2𝑜 ∧ (𝑇𝑦) = 𝐹) → ((𝐷 ∈ V ∧ 𝑃𝐷𝑃 ≈ 2𝑜) ∧ 𝐹 = (𝑇𝑃))))
5423, 53sylbid 231 . 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 197  wa 384  w3a 1100   = wceq 1637  wcel 2156  wral 3096  wrex 3097  {crab 3100  Vcvv 3391  cdif 3766  wss 3769  c0 4116  ifcif 4279  𝒫 cpw 4351  {csn 4370   cuni 4630   class class class wbr 4844  cmpt 4923   I cid 5218  dom cdm 5311  ran crn 5312   Fn wfn 6096  cfv 6101  2𝑜c2o 7790  cen 8189  pmTrspcpmtr 18062
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2068  ax-7 2104  ax-8 2158  ax-9 2165  ax-10 2185  ax-11 2201  ax-12 2214  ax-13 2420  ax-ext 2784  ax-rep 4964  ax-sep 4975  ax-nul 4983  ax-pow 5035  ax-pr 5096  ax-un 7179
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3or 1101  df-3an 1102  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2061  df-eu 2634  df-mo 2635  df-clab 2793  df-cleq 2799  df-clel 2802  df-nfc 2937  df-ne 2979  df-ral 3101  df-rex 3102  df-reu 3103  df-rab 3105  df-v 3393  df-sbc 3634  df-csb 3729  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-pss 3785  df-nul 4117  df-if 4280  df-pw 4353  df-sn 4371  df-pr 4373  df-tp 4375  df-op 4377  df-uni 4631  df-iun 4714  df-br 4845  df-opab 4907  df-mpt 4924  df-tr 4947  df-id 5219  df-eprel 5224  df-po 5232  df-so 5233  df-fr 5270  df-we 5272  df-xp 5317  df-rel 5318  df-cnv 5319  df-co 5320  df-dm 5321  df-rn 5322  df-res 5323  df-ima 5324  df-ord 5939  df-on 5940  df-lim 5941  df-suc 5942  df-iota 6064  df-fun 6103  df-fn 6104  df-f 6105  df-f1 6106  df-fo 6107  df-f1o 6108  df-fv 6109  df-om 7296  df-1o 7796  df-2o 7797  df-er 7979  df-en 8193  df-fin 8196  df-pmtr 18063
This theorem is referenced by:  pmtrffv  18080  pmtrrn2  18081  pmtrfinv  18082  pmtrfmvdn0  18083  pmtrff1o  18084  pmtrfcnv  18085  pmtrfb  18086
  Copyright terms: Public domain W3C validator