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

Theorem pmtrfrn 18098
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 4062 . . . 4 ¬ 𝐹 ∈ ∅
2 pmtrrn.r . . . . . 6 𝑅 = ran 𝑇
3 pmtrrn.t . . . . . . . . 9 𝑇 = (pmTrsp‘𝐷)
4 fvprc 6347 . . . . . . . . 9 𝐷 ∈ V → (pmTrsp‘𝐷) = ∅)
53, 4syl5eq 2806 . . . . . . . 8 𝐷 ∈ V → 𝑇 = ∅)
65rneqd 5508 . . . . . . 7 𝐷 ∈ V → ran 𝑇 = ran ∅)
7 rn0 5532 . . . . . . 7 ran ∅ = ∅
86, 7syl6eq 2810 . . . . . 6 𝐷 ∈ V → ran 𝑇 = ∅)
92, 8syl5eq 2806 . . . . 5 𝐷 ∈ V → 𝑅 = ∅)
109eleq2d 2825 . . . 4 𝐷 ∈ V → (𝐹𝑅𝐹 ∈ ∅))
111, 10mtbiri 316 . . 3 𝐷 ∈ V → ¬ 𝐹𝑅)
1211con4i 113 . 2 (𝐹𝑅𝐷 ∈ V)
13 mptexg 6649 . . . . . . . 8 (𝐷 ∈ V → (𝑧𝐷 ↦ if(𝑧𝑤, (𝑤 ∖ {𝑧}), 𝑧)) ∈ V)
1413ralrimivw 3105 . . . . . . 7 (𝐷 ∈ V → ∀𝑤 ∈ {𝑥 ∈ 𝒫 𝐷𝑥 ≈ 2𝑜} (𝑧𝐷 ↦ if(𝑧𝑤, (𝑤 ∖ {𝑧}), 𝑧)) ∈ V)
15 eqid 2760 . . . . . . . 8 (𝑤 ∈ {𝑥 ∈ 𝒫 𝐷𝑥 ≈ 2𝑜} ↦ (𝑧𝐷 ↦ if(𝑧𝑤, (𝑤 ∖ {𝑧}), 𝑧))) = (𝑤 ∈ {𝑥 ∈ 𝒫 𝐷𝑥 ≈ 2𝑜} ↦ (𝑧𝐷 ↦ if(𝑧𝑤, (𝑤 ∖ {𝑧}), 𝑧)))
1615fnmpt 6181 . . . . . . 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 18090 . . . . . . 7 (𝐷 ∈ V → 𝑇 = (𝑤 ∈ {𝑥 ∈ 𝒫 𝐷𝑥 ≈ 2𝑜} ↦ (𝑧𝐷 ↦ if(𝑧𝑤, (𝑤 ∖ {𝑧}), 𝑧))))
1918fneq1d 6142 . . . . . 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 6406 . . . . 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 2831 . . . 4 (𝐹𝑅𝐹 ∈ ran 𝑇)
24 breq1 4807 . . . . . 6 (𝑥 = 𝑦 → (𝑥 ≈ 2𝑜𝑦 ≈ 2𝑜))
2524rexrab 3511 . . . . 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 4312 . . . . 5 (𝑦 ∈ 𝒫 𝐷𝑦𝐷)
29 simp1 1131 . . . . . . . . . 10 ((𝐷 ∈ V ∧ 𝑦𝐷𝑦 ≈ 2𝑜) → 𝐷 ∈ V)
303pmtrmvd 18096 . . . . . . . . . . 11 ((𝐷 ∈ V ∧ 𝑦𝐷𝑦 ≈ 2𝑜) → dom ((𝑇𝑦) ∖ I ) = 𝑦)
31 simp2 1132 . . . . . . . . . . 11 ((𝐷 ∈ V ∧ 𝑦𝐷𝑦 ≈ 2𝑜) → 𝑦𝐷)
3230, 31eqsstrd 3780 . . . . . . . . . 10 ((𝐷 ∈ V ∧ 𝑦𝐷𝑦 ≈ 2𝑜) → dom ((𝑇𝑦) ∖ I ) ⊆ 𝐷)
33 simp3 1133 . . . . . . . . . . 11 ((𝐷 ∈ V ∧ 𝑦𝐷𝑦 ≈ 2𝑜) → 𝑦 ≈ 2𝑜)
3430, 33eqbrtrd 4826 . . . . . . . . . 10 ((𝐷 ∈ V ∧ 𝑦𝐷𝑦 ≈ 2𝑜) → dom ((𝑇𝑦) ∖ I ) ≈ 2𝑜)
3529, 32, 343jca 1123 . . . . . . . . 9 ((𝐷 ∈ V ∧ 𝑦𝐷𝑦 ≈ 2𝑜) → (𝐷 ∈ V ∧ dom ((𝑇𝑦) ∖ I ) ⊆ 𝐷 ∧ dom ((𝑇𝑦) ∖ I ) ≈ 2𝑜))
3630eqcomd 2766 . . . . . . . . . 10 ((𝐷 ∈ V ∧ 𝑦𝐷𝑦 ≈ 2𝑜) → 𝑦 = dom ((𝑇𝑦) ∖ I ))
3736fveq2d 6357 . . . . . . . . 9 ((𝐷 ∈ V ∧ 𝑦𝐷𝑦 ≈ 2𝑜) → (𝑇𝑦) = (𝑇‘dom ((𝑇𝑦) ∖ I )))
3835, 37jca 555 . . . . . . . 8 ((𝐷 ∈ V ∧ 𝑦𝐷𝑦 ≈ 2𝑜) → ((𝐷 ∈ V ∧ dom ((𝑇𝑦) ∖ I ) ⊆ 𝐷 ∧ dom ((𝑇𝑦) ∖ I ) ≈ 2𝑜) ∧ (𝑇𝑦) = (𝑇‘dom ((𝑇𝑦) ∖ I ))))
39 difeq1 3864 . . . . . . . . . . 11 ((𝑇𝑦) = 𝐹 → ((𝑇𝑦) ∖ I ) = (𝐹 ∖ I ))
4039dmeqd 5481 . . . . . . . . . 10 ((𝑇𝑦) = 𝐹 → dom ((𝑇𝑦) ∖ I ) = dom (𝐹 ∖ I ))
41 pmtrfrn.p . . . . . . . . . 10 𝑃 = dom (𝐹 ∖ I )
4240, 41syl6eqr 2812 . . . . . . . . 9 ((𝑇𝑦) = 𝐹 → dom ((𝑇𝑦) ∖ I ) = 𝑃)
43 sseq1 3767 . . . . . . . . . . . 12 (dom ((𝑇𝑦) ∖ I ) = 𝑃 → (dom ((𝑇𝑦) ∖ I ) ⊆ 𝐷𝑃𝐷))
44 breq1 4807 . . . . . . . . . . . 12 (dom ((𝑇𝑦) ∖ I ) = 𝑃 → (dom ((𝑇𝑦) ∖ I ) ≈ 2𝑜𝑃 ≈ 2𝑜))
4543, 443anbi23d 1551 . . . . . . . . . . 11 (dom ((𝑇𝑦) ∖ I ) = 𝑃 → ((𝐷 ∈ V ∧ dom ((𝑇𝑦) ∖ I ) ⊆ 𝐷 ∧ dom ((𝑇𝑦) ∖ I ) ≈ 2𝑜) ↔ (𝐷 ∈ V ∧ 𝑃𝐷𝑃 ≈ 2𝑜)))
4645adantl 473 . . . . . . . . . 10 (((𝑇𝑦) = 𝐹 ∧ dom ((𝑇𝑦) ∖ I ) = 𝑃) → ((𝐷 ∈ V ∧ dom ((𝑇𝑦) ∖ I ) ⊆ 𝐷 ∧ dom ((𝑇𝑦) ∖ I ) ≈ 2𝑜) ↔ (𝐷 ∈ V ∧ 𝑃𝐷𝑃 ≈ 2𝑜)))
47 simpl 474 . . . . . . . . . . 11 (((𝑇𝑦) = 𝐹 ∧ dom ((𝑇𝑦) ∖ I ) = 𝑃) → (𝑇𝑦) = 𝐹)
48 fveq2 6353 . . . . . . . . . . . 12 (dom ((𝑇𝑦) ∖ I ) = 𝑃 → (𝑇‘dom ((𝑇𝑦) ∖ I )) = (𝑇𝑃))
4948adantl 473 . . . . . . . . . . 11 (((𝑇𝑦) = 𝐹 ∧ dom ((𝑇𝑦) ∖ I ) = 𝑃) → (𝑇‘dom ((𝑇𝑦) ∖ I )) = (𝑇𝑃))
5047, 49eqeq12d 2775 . . . . . . . . . 10 (((𝑇𝑦) = 𝐹 ∧ dom ((𝑇𝑦) ∖ I ) = 𝑃) → ((𝑇𝑦) = (𝑇‘dom ((𝑇𝑦) ∖ I )) ↔ 𝐹 = (𝑇𝑃)))
5146, 50anbi12d 749 . . . . . . . . 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 705 . . . . . . . 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 1113 . . . . . 6 (𝐷 ∈ V → (𝑦𝐷 → (𝑦 ≈ 2𝑜 → ((𝑇𝑦) = 𝐹 → ((𝐷 ∈ V ∧ 𝑃𝐷𝑃 ≈ 2𝑜) ∧ 𝐹 = (𝑇𝑃))))))
5554imp4a 615 . . . . 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 3168 . . 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 1072   = wceq 1632  wcel 2139  wral 3050  wrex 3051  {crab 3054  Vcvv 3340  cdif 3712  wss 3715  c0 4058  ifcif 4230  𝒫 cpw 4302  {csn 4321   cuni 4588   class class class wbr 4804  cmpt 4881   I cid 5173  dom cdm 5266  ran crn 5267   Fn wfn 6044  cfv 6049  2𝑜c2o 7724  cen 8120  pmTrspcpmtr 18081
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7115
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-reu 3057  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-pss 3731  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-tp 4326  df-op 4328  df-uni 4589  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-tr 4905  df-id 5174  df-eprel 5179  df-po 5187  df-so 5188  df-fr 5225  df-we 5227  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-ord 5887  df-on 5888  df-lim 5889  df-suc 5890  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-om 7232  df-1o 7730  df-2o 7731  df-er 7913  df-en 8124  df-fin 8127  df-pmtr 18082
This theorem is referenced by:  pmtrffv  18099  pmtrrn2  18100  pmtrfinv  18101  pmtrfmvdn0  18102  pmtrff1o  18103  pmtrfcnv  18104  pmtrfb  18105
  Copyright terms: Public domain W3C validator