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

Theorem pmtrfinv 17805
Description: A transposition function is an involution. (Contributed by Stefan O'Rear, 22-Aug-2015.)
Hypotheses
Ref Expression
pmtrrn.t 𝑇 = (pmTrsp‘𝐷)
pmtrrn.r 𝑅 = ran 𝑇
Assertion
Ref Expression
pmtrfinv (𝐹𝑅 → (𝐹𝐹) = ( I ↾ 𝐷))

Proof of Theorem pmtrfinv
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 pmtrrn.t . . . . . . 7 𝑇 = (pmTrsp‘𝐷)
2 pmtrrn.r . . . . . . 7 𝑅 = ran 𝑇
3 eqid 2621 . . . . . . 7 dom (𝐹 ∖ I ) = dom (𝐹 ∖ I )
41, 2, 3pmtrfrn 17802 . . . . . 6 (𝐹𝑅 → ((𝐷 ∈ V ∧ dom (𝐹 ∖ I ) ⊆ 𝐷 ∧ dom (𝐹 ∖ I ) ≈ 2𝑜) ∧ 𝐹 = (𝑇‘dom (𝐹 ∖ I ))))
54simpld 475 . . . . 5 (𝐹𝑅 → (𝐷 ∈ V ∧ dom (𝐹 ∖ I ) ⊆ 𝐷 ∧ dom (𝐹 ∖ I ) ≈ 2𝑜))
61pmtrf 17799 . . . . 5 ((𝐷 ∈ V ∧ dom (𝐹 ∖ I ) ⊆ 𝐷 ∧ dom (𝐹 ∖ I ) ≈ 2𝑜) → (𝑇‘dom (𝐹 ∖ I )):𝐷𝐷)
75, 6syl 17 . . . 4 (𝐹𝑅 → (𝑇‘dom (𝐹 ∖ I )):𝐷𝐷)
84simprd 479 . . . . 5 (𝐹𝑅𝐹 = (𝑇‘dom (𝐹 ∖ I )))
98feq1d 5989 . . . 4 (𝐹𝑅 → (𝐹:𝐷𝐷 ↔ (𝑇‘dom (𝐹 ∖ I )):𝐷𝐷))
107, 9mpbird 247 . . 3 (𝐹𝑅𝐹:𝐷𝐷)
11 fco 6017 . . . 4 ((𝐹:𝐷𝐷𝐹:𝐷𝐷) → (𝐹𝐹):𝐷𝐷)
1211anidms 676 . . 3 (𝐹:𝐷𝐷 → (𝐹𝐹):𝐷𝐷)
13 ffn 6004 . . 3 ((𝐹𝐹):𝐷𝐷 → (𝐹𝐹) Fn 𝐷)
1410, 12, 133syl 18 . 2 (𝐹𝑅 → (𝐹𝐹) Fn 𝐷)
15 fnresi 5968 . . 3 ( I ↾ 𝐷) Fn 𝐷
1615a1i 11 . 2 (𝐹𝑅 → ( I ↾ 𝐷) Fn 𝐷)
171, 2, 3pmtrffv 17803 . . . . . . 7 ((𝐹𝑅𝑥𝐷) → (𝐹𝑥) = if(𝑥 ∈ dom (𝐹 ∖ I ), (dom (𝐹 ∖ I ) ∖ {𝑥}), 𝑥))
18 iftrue 4066 . . . . . . 7 (𝑥 ∈ dom (𝐹 ∖ I ) → if(𝑥 ∈ dom (𝐹 ∖ I ), (dom (𝐹 ∖ I ) ∖ {𝑥}), 𝑥) = (dom (𝐹 ∖ I ) ∖ {𝑥}))
1917, 18sylan9eq 2675 . . . . . 6 (((𝐹𝑅𝑥𝐷) ∧ 𝑥 ∈ dom (𝐹 ∖ I )) → (𝐹𝑥) = (dom (𝐹 ∖ I ) ∖ {𝑥}))
2019fveq2d 6154 . . . . 5 (((𝐹𝑅𝑥𝐷) ∧ 𝑥 ∈ dom (𝐹 ∖ I )) → (𝐹‘(𝐹𝑥)) = (𝐹 (dom (𝐹 ∖ I ) ∖ {𝑥})))
21 simpll 789 . . . . . . 7 (((𝐹𝑅𝑥𝐷) ∧ 𝑥 ∈ dom (𝐹 ∖ I )) → 𝐹𝑅)
225simp2d 1072 . . . . . . . . 9 (𝐹𝑅 → dom (𝐹 ∖ I ) ⊆ 𝐷)
2322ad2antrr 761 . . . . . . . 8 (((𝐹𝑅𝑥𝐷) ∧ 𝑥 ∈ dom (𝐹 ∖ I )) → dom (𝐹 ∖ I ) ⊆ 𝐷)
24 1onn 7667 . . . . . . . . . . . 12 1𝑜 ∈ ω
2524a1i 11 . . . . . . . . . . 11 (((𝐹𝑅𝑥𝐷) ∧ 𝑥 ∈ dom (𝐹 ∖ I )) → 1𝑜 ∈ ω)
265simp3d 1073 . . . . . . . . . . . . 13 (𝐹𝑅 → dom (𝐹 ∖ I ) ≈ 2𝑜)
27 df-2o 7509 . . . . . . . . . . . . 13 2𝑜 = suc 1𝑜
2826, 27syl6breq 4656 . . . . . . . . . . . 12 (𝐹𝑅 → dom (𝐹 ∖ I ) ≈ suc 1𝑜)
2928ad2antrr 761 . . . . . . . . . . 11 (((𝐹𝑅𝑥𝐷) ∧ 𝑥 ∈ dom (𝐹 ∖ I )) → dom (𝐹 ∖ I ) ≈ suc 1𝑜)
30 simpr 477 . . . . . . . . . . 11 (((𝐹𝑅𝑥𝐷) ∧ 𝑥 ∈ dom (𝐹 ∖ I )) → 𝑥 ∈ dom (𝐹 ∖ I ))
31 dif1en 8140 . . . . . . . . . . 11 ((1𝑜 ∈ ω ∧ dom (𝐹 ∖ I ) ≈ suc 1𝑜𝑥 ∈ dom (𝐹 ∖ I )) → (dom (𝐹 ∖ I ) ∖ {𝑥}) ≈ 1𝑜)
3225, 29, 30, 31syl3anc 1323 . . . . . . . . . 10 (((𝐹𝑅𝑥𝐷) ∧ 𝑥 ∈ dom (𝐹 ∖ I )) → (dom (𝐹 ∖ I ) ∖ {𝑥}) ≈ 1𝑜)
33 en1uniel 7975 . . . . . . . . . 10 ((dom (𝐹 ∖ I ) ∖ {𝑥}) ≈ 1𝑜 (dom (𝐹 ∖ I ) ∖ {𝑥}) ∈ (dom (𝐹 ∖ I ) ∖ {𝑥}))
3432, 33syl 17 . . . . . . . . 9 (((𝐹𝑅𝑥𝐷) ∧ 𝑥 ∈ dom (𝐹 ∖ I )) → (dom (𝐹 ∖ I ) ∖ {𝑥}) ∈ (dom (𝐹 ∖ I ) ∖ {𝑥}))
3534eldifad 3568 . . . . . . . 8 (((𝐹𝑅𝑥𝐷) ∧ 𝑥 ∈ dom (𝐹 ∖ I )) → (dom (𝐹 ∖ I ) ∖ {𝑥}) ∈ dom (𝐹 ∖ I ))
3623, 35sseldd 3585 . . . . . . 7 (((𝐹𝑅𝑥𝐷) ∧ 𝑥 ∈ dom (𝐹 ∖ I )) → (dom (𝐹 ∖ I ) ∖ {𝑥}) ∈ 𝐷)
371, 2, 3pmtrffv 17803 . . . . . . 7 ((𝐹𝑅 (dom (𝐹 ∖ I ) ∖ {𝑥}) ∈ 𝐷) → (𝐹 (dom (𝐹 ∖ I ) ∖ {𝑥})) = if( (dom (𝐹 ∖ I ) ∖ {𝑥}) ∈ dom (𝐹 ∖ I ), (dom (𝐹 ∖ I ) ∖ { (dom (𝐹 ∖ I ) ∖ {𝑥})}), (dom (𝐹 ∖ I ) ∖ {𝑥})))
3821, 36, 37syl2anc 692 . . . . . 6 (((𝐹𝑅𝑥𝐷) ∧ 𝑥 ∈ dom (𝐹 ∖ I )) → (𝐹 (dom (𝐹 ∖ I ) ∖ {𝑥})) = if( (dom (𝐹 ∖ I ) ∖ {𝑥}) ∈ dom (𝐹 ∖ I ), (dom (𝐹 ∖ I ) ∖ { (dom (𝐹 ∖ I ) ∖ {𝑥})}), (dom (𝐹 ∖ I ) ∖ {𝑥})))
39 iftrue 4066 . . . . . . . 8 ( (dom (𝐹 ∖ I ) ∖ {𝑥}) ∈ dom (𝐹 ∖ I ) → if( (dom (𝐹 ∖ I ) ∖ {𝑥}) ∈ dom (𝐹 ∖ I ), (dom (𝐹 ∖ I ) ∖ { (dom (𝐹 ∖ I ) ∖ {𝑥})}), (dom (𝐹 ∖ I ) ∖ {𝑥})) = (dom (𝐹 ∖ I ) ∖ { (dom (𝐹 ∖ I ) ∖ {𝑥})}))
4035, 39syl 17 . . . . . . 7 (((𝐹𝑅𝑥𝐷) ∧ 𝑥 ∈ dom (𝐹 ∖ I )) → if( (dom (𝐹 ∖ I ) ∖ {𝑥}) ∈ dom (𝐹 ∖ I ), (dom (𝐹 ∖ I ) ∖ { (dom (𝐹 ∖ I ) ∖ {𝑥})}), (dom (𝐹 ∖ I ) ∖ {𝑥})) = (dom (𝐹 ∖ I ) ∖ { (dom (𝐹 ∖ I ) ∖ {𝑥})}))
4126adantr 481 . . . . . . . 8 ((𝐹𝑅𝑥𝐷) → dom (𝐹 ∖ I ) ≈ 2𝑜)
42 en2other2 8779 . . . . . . . . 9 ((𝑥 ∈ dom (𝐹 ∖ I ) ∧ dom (𝐹 ∖ I ) ≈ 2𝑜) → (dom (𝐹 ∖ I ) ∖ { (dom (𝐹 ∖ I ) ∖ {𝑥})}) = 𝑥)
4342ancoms 469 . . . . . . . 8 ((dom (𝐹 ∖ I ) ≈ 2𝑜𝑥 ∈ dom (𝐹 ∖ I )) → (dom (𝐹 ∖ I ) ∖ { (dom (𝐹 ∖ I ) ∖ {𝑥})}) = 𝑥)
4441, 43sylan 488 . . . . . . 7 (((𝐹𝑅𝑥𝐷) ∧ 𝑥 ∈ dom (𝐹 ∖ I )) → (dom (𝐹 ∖ I ) ∖ { (dom (𝐹 ∖ I ) ∖ {𝑥})}) = 𝑥)
4540, 44eqtrd 2655 . . . . . 6 (((𝐹𝑅𝑥𝐷) ∧ 𝑥 ∈ dom (𝐹 ∖ I )) → if( (dom (𝐹 ∖ I ) ∖ {𝑥}) ∈ dom (𝐹 ∖ I ), (dom (𝐹 ∖ I ) ∖ { (dom (𝐹 ∖ I ) ∖ {𝑥})}), (dom (𝐹 ∖ I ) ∖ {𝑥})) = 𝑥)
4638, 45eqtrd 2655 . . . . 5 (((𝐹𝑅𝑥𝐷) ∧ 𝑥 ∈ dom (𝐹 ∖ I )) → (𝐹 (dom (𝐹 ∖ I ) ∖ {𝑥})) = 𝑥)
4720, 46eqtrd 2655 . . . 4 (((𝐹𝑅𝑥𝐷) ∧ 𝑥 ∈ dom (𝐹 ∖ I )) → (𝐹‘(𝐹𝑥)) = 𝑥)
48 ffn 6004 . . . . . . . . 9 (𝐹:𝐷𝐷𝐹 Fn 𝐷)
4910, 48syl 17 . . . . . . . 8 (𝐹𝑅𝐹 Fn 𝐷)
50 fnelnfp 6400 . . . . . . . 8 ((𝐹 Fn 𝐷𝑥𝐷) → (𝑥 ∈ dom (𝐹 ∖ I ) ↔ (𝐹𝑥) ≠ 𝑥))
5149, 50sylan 488 . . . . . . 7 ((𝐹𝑅𝑥𝐷) → (𝑥 ∈ dom (𝐹 ∖ I ) ↔ (𝐹𝑥) ≠ 𝑥))
5251necon2bbid 2833 . . . . . 6 ((𝐹𝑅𝑥𝐷) → ((𝐹𝑥) = 𝑥 ↔ ¬ 𝑥 ∈ dom (𝐹 ∖ I )))
5352biimpar 502 . . . . 5 (((𝐹𝑅𝑥𝐷) ∧ ¬ 𝑥 ∈ dom (𝐹 ∖ I )) → (𝐹𝑥) = 𝑥)
54 fveq2 6150 . . . . . 6 ((𝐹𝑥) = 𝑥 → (𝐹‘(𝐹𝑥)) = (𝐹𝑥))
55 id 22 . . . . . 6 ((𝐹𝑥) = 𝑥 → (𝐹𝑥) = 𝑥)
5654, 55eqtrd 2655 . . . . 5 ((𝐹𝑥) = 𝑥 → (𝐹‘(𝐹𝑥)) = 𝑥)
5753, 56syl 17 . . . 4 (((𝐹𝑅𝑥𝐷) ∧ ¬ 𝑥 ∈ dom (𝐹 ∖ I )) → (𝐹‘(𝐹𝑥)) = 𝑥)
5847, 57pm2.61dan 831 . . 3 ((𝐹𝑅𝑥𝐷) → (𝐹‘(𝐹𝑥)) = 𝑥)
59 fvco2 6232 . . . 4 ((𝐹 Fn 𝐷𝑥𝐷) → ((𝐹𝐹)‘𝑥) = (𝐹‘(𝐹𝑥)))
6049, 59sylan 488 . . 3 ((𝐹𝑅𝑥𝐷) → ((𝐹𝐹)‘𝑥) = (𝐹‘(𝐹𝑥)))
61 fvresi 6396 . . . 4 (𝑥𝐷 → (( I ↾ 𝐷)‘𝑥) = 𝑥)
6261adantl 482 . . 3 ((𝐹𝑅𝑥𝐷) → (( I ↾ 𝐷)‘𝑥) = 𝑥)
6358, 60, 623eqtr4d 2665 . 2 ((𝐹𝑅𝑥𝐷) → ((𝐹𝐹)‘𝑥) = (( I ↾ 𝐷)‘𝑥))
6414, 16, 63eqfnfvd 6272 1 (𝐹𝑅 → (𝐹𝐹) = ( I ↾ 𝐷))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 384  w3a 1036   = wceq 1480  wcel 1987  wne 2790  Vcvv 3186  cdif 3553  wss 3556  ifcif 4060  {csn 4150   cuni 4404   class class class wbr 4615   I cid 4986  dom cdm 5076  ran crn 5077  cres 5078  ccom 5080  suc csuc 5686   Fn wfn 5844  wf 5845  cfv 5849  ωcom 7015  1𝑜c1o 7501  2𝑜c2o 7502  cen 7899  pmTrspcpmtr 17785
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4733  ax-sep 4743  ax-nul 4751  ax-pow 4805  ax-pr 4869  ax-un 6905
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3188  df-sbc 3419  df-csb 3516  df-dif 3559  df-un 3561  df-in 3563  df-ss 3570  df-pss 3572  df-nul 3894  df-if 4061  df-pw 4134  df-sn 4151  df-pr 4153  df-tp 4155  df-op 4157  df-uni 4405  df-iun 4489  df-br 4616  df-opab 4676  df-mpt 4677  df-tr 4715  df-eprel 4987  df-id 4991  df-po 4997  df-so 4998  df-fr 5035  df-we 5037  df-xp 5082  df-rel 5083  df-cnv 5084  df-co 5085  df-dm 5086  df-rn 5087  df-res 5088  df-ima 5089  df-ord 5687  df-on 5688  df-lim 5689  df-suc 5690  df-iota 5812  df-fun 5851  df-fn 5852  df-f 5853  df-f1 5854  df-fo 5855  df-f1o 5856  df-fv 5857  df-om 7016  df-1o 7508  df-2o 7509  df-er 7690  df-en 7903  df-dom 7904  df-sdom 7905  df-fin 7906  df-pmtr 17786
This theorem is referenced by:  pmtrff1o  17807  pmtrfcnv  17808  symggen  17814  psgnunilem1  17837
  Copyright terms: Public domain W3C validator