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Theorem pmtrfinv 19370
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 2730 . . . . . . 7 dom (𝐹 ∖ I ) = dom (𝐹 ∖ I )
41, 2, 3pmtrfrn 19367 . . . . . 6 (𝐹𝑅 → ((𝐷 ∈ V ∧ dom (𝐹 ∖ I ) ⊆ 𝐷 ∧ dom (𝐹 ∖ I ) ≈ 2o) ∧ 𝐹 = (𝑇‘dom (𝐹 ∖ I ))))
54simpld 493 . . . . 5 (𝐹𝑅 → (𝐷 ∈ V ∧ dom (𝐹 ∖ I ) ⊆ 𝐷 ∧ dom (𝐹 ∖ I ) ≈ 2o))
61pmtrf 19364 . . . . 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 494 . . . . 5 (𝐹𝑅𝐹 = (𝑇‘dom (𝐹 ∖ I )))
98feq1d 6701 . . . 4 (𝐹𝑅 → (𝐹:𝐷𝐷 ↔ (𝑇‘dom (𝐹 ∖ I )):𝐷𝐷))
107, 9mpbird 256 . . 3 (𝐹𝑅𝐹:𝐷𝐷)
11 fco 6740 . . . 4 ((𝐹:𝐷𝐷𝐹:𝐷𝐷) → (𝐹𝐹):𝐷𝐷)
1211anidms 565 . . 3 (𝐹:𝐷𝐷 → (𝐹𝐹):𝐷𝐷)
13 ffn 6716 . . 3 ((𝐹𝐹):𝐷𝐷 → (𝐹𝐹) Fn 𝐷)
1410, 12, 133syl 18 . 2 (𝐹𝑅 → (𝐹𝐹) Fn 𝐷)
15 fnresi 6678 . . 3 ( I ↾ 𝐷) Fn 𝐷
1615a1i 11 . 2 (𝐹𝑅 → ( I ↾ 𝐷) Fn 𝐷)
171, 2, 3pmtrffv 19368 . . . . . . 7 ((𝐹𝑅𝑥𝐷) → (𝐹𝑥) = if(𝑥 ∈ dom (𝐹 ∖ I ), (dom (𝐹 ∖ I ) ∖ {𝑥}), 𝑥))
18 iftrue 4533 . . . . . . 7 (𝑥 ∈ dom (𝐹 ∖ I ) → if(𝑥 ∈ dom (𝐹 ∖ I ), (dom (𝐹 ∖ I ) ∖ {𝑥}), 𝑥) = (dom (𝐹 ∖ I ) ∖ {𝑥}))
1917, 18sylan9eq 2790 . . . . . 6 (((𝐹𝑅𝑥𝐷) ∧ 𝑥 ∈ dom (𝐹 ∖ I )) → (𝐹𝑥) = (dom (𝐹 ∖ I ) ∖ {𝑥}))
2019fveq2d 6894 . . . . 5 (((𝐹𝑅𝑥𝐷) ∧ 𝑥 ∈ dom (𝐹 ∖ I )) → (𝐹‘(𝐹𝑥)) = (𝐹 (dom (𝐹 ∖ I ) ∖ {𝑥})))
21 simpll 763 . . . . . . 7 (((𝐹𝑅𝑥𝐷) ∧ 𝑥 ∈ dom (𝐹 ∖ I )) → 𝐹𝑅)
225simp2d 1141 . . . . . . . . 9 (𝐹𝑅 → dom (𝐹 ∖ I ) ⊆ 𝐷)
2322ad2antrr 722 . . . . . . . 8 (((𝐹𝑅𝑥𝐷) ∧ 𝑥 ∈ dom (𝐹 ∖ I )) → dom (𝐹 ∖ I ) ⊆ 𝐷)
24 1onn 8641 . . . . . . . . . . 11 1o ∈ ω
255simp3d 1142 . . . . . . . . . . . . 13 (𝐹𝑅 → dom (𝐹 ∖ I ) ≈ 2o)
26 df-2o 8469 . . . . . . . . . . . . 13 2o = suc 1o
2725, 26breqtrdi 5188 . . . . . . . . . . . 12 (𝐹𝑅 → dom (𝐹 ∖ I ) ≈ suc 1o)
2827ad2antrr 722 . . . . . . . . . . 11 (((𝐹𝑅𝑥𝐷) ∧ 𝑥 ∈ dom (𝐹 ∖ I )) → dom (𝐹 ∖ I ) ≈ suc 1o)
29 simpr 483 . . . . . . . . . . 11 (((𝐹𝑅𝑥𝐷) ∧ 𝑥 ∈ dom (𝐹 ∖ I )) → 𝑥 ∈ dom (𝐹 ∖ I ))
30 dif1ennn 9163 . . . . . . . . . . 11 ((1o ∈ ω ∧ dom (𝐹 ∖ I ) ≈ suc 1o𝑥 ∈ dom (𝐹 ∖ I )) → (dom (𝐹 ∖ I ) ∖ {𝑥}) ≈ 1o)
3124, 28, 29, 30mp3an2i 1464 . . . . . . . . . 10 (((𝐹𝑅𝑥𝐷) ∧ 𝑥 ∈ dom (𝐹 ∖ I )) → (dom (𝐹 ∖ I ) ∖ {𝑥}) ≈ 1o)
32 en1uniel 9030 . . . . . . . . . 10 ((dom (𝐹 ∖ I ) ∖ {𝑥}) ≈ 1o (dom (𝐹 ∖ I ) ∖ {𝑥}) ∈ (dom (𝐹 ∖ I ) ∖ {𝑥}))
3331, 32syl 17 . . . . . . . . 9 (((𝐹𝑅𝑥𝐷) ∧ 𝑥 ∈ dom (𝐹 ∖ I )) → (dom (𝐹 ∖ I ) ∖ {𝑥}) ∈ (dom (𝐹 ∖ I ) ∖ {𝑥}))
3433eldifad 3959 . . . . . . . 8 (((𝐹𝑅𝑥𝐷) ∧ 𝑥 ∈ dom (𝐹 ∖ I )) → (dom (𝐹 ∖ I ) ∖ {𝑥}) ∈ dom (𝐹 ∖ I ))
3523, 34sseldd 3982 . . . . . . 7 (((𝐹𝑅𝑥𝐷) ∧ 𝑥 ∈ dom (𝐹 ∖ I )) → (dom (𝐹 ∖ I ) ∖ {𝑥}) ∈ 𝐷)
361, 2, 3pmtrffv 19368 . . . . . . 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})))
3721, 35, 36syl2anc 582 . . . . . 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})))
38 iftrue 4533 . . . . . . . 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})}))
3934, 38syl 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})}))
4025adantr 479 . . . . . . . 8 ((𝐹𝑅𝑥𝐷) → dom (𝐹 ∖ I ) ≈ 2o)
41 en2other2 10006 . . . . . . . . 9 ((𝑥 ∈ dom (𝐹 ∖ I ) ∧ dom (𝐹 ∖ I ) ≈ 2o) → (dom (𝐹 ∖ I ) ∖ { (dom (𝐹 ∖ I ) ∖ {𝑥})}) = 𝑥)
4241ancoms 457 . . . . . . . 8 ((dom (𝐹 ∖ I ) ≈ 2o𝑥 ∈ dom (𝐹 ∖ I )) → (dom (𝐹 ∖ I ) ∖ { (dom (𝐹 ∖ I ) ∖ {𝑥})}) = 𝑥)
4340, 42sylan 578 . . . . . . 7 (((𝐹𝑅𝑥𝐷) ∧ 𝑥 ∈ dom (𝐹 ∖ I )) → (dom (𝐹 ∖ I ) ∖ { (dom (𝐹 ∖ I ) ∖ {𝑥})}) = 𝑥)
4439, 43eqtrd 2770 . . . . . 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)
4537, 44eqtrd 2770 . . . . 5 (((𝐹𝑅𝑥𝐷) ∧ 𝑥 ∈ dom (𝐹 ∖ I )) → (𝐹 (dom (𝐹 ∖ I ) ∖ {𝑥})) = 𝑥)
4620, 45eqtrd 2770 . . . 4 (((𝐹𝑅𝑥𝐷) ∧ 𝑥 ∈ dom (𝐹 ∖ I )) → (𝐹‘(𝐹𝑥)) = 𝑥)
4710ffnd 6717 . . . . . . . 8 (𝐹𝑅𝐹 Fn 𝐷)
48 fnelnfp 7176 . . . . . . . 8 ((𝐹 Fn 𝐷𝑥𝐷) → (𝑥 ∈ dom (𝐹 ∖ I ) ↔ (𝐹𝑥) ≠ 𝑥))
4947, 48sylan 578 . . . . . . 7 ((𝐹𝑅𝑥𝐷) → (𝑥 ∈ dom (𝐹 ∖ I ) ↔ (𝐹𝑥) ≠ 𝑥))
5049necon2bbid 2982 . . . . . 6 ((𝐹𝑅𝑥𝐷) → ((𝐹𝑥) = 𝑥 ↔ ¬ 𝑥 ∈ dom (𝐹 ∖ I )))
5150biimpar 476 . . . . 5 (((𝐹𝑅𝑥𝐷) ∧ ¬ 𝑥 ∈ dom (𝐹 ∖ I )) → (𝐹𝑥) = 𝑥)
52 fveq2 6890 . . . . . 6 ((𝐹𝑥) = 𝑥 → (𝐹‘(𝐹𝑥)) = (𝐹𝑥))
53 id 22 . . . . . 6 ((𝐹𝑥) = 𝑥 → (𝐹𝑥) = 𝑥)
5452, 53eqtrd 2770 . . . . 5 ((𝐹𝑥) = 𝑥 → (𝐹‘(𝐹𝑥)) = 𝑥)
5551, 54syl 17 . . . 4 (((𝐹𝑅𝑥𝐷) ∧ ¬ 𝑥 ∈ dom (𝐹 ∖ I )) → (𝐹‘(𝐹𝑥)) = 𝑥)
5646, 55pm2.61dan 809 . . 3 ((𝐹𝑅𝑥𝐷) → (𝐹‘(𝐹𝑥)) = 𝑥)
57 fvco2 6987 . . . 4 ((𝐹 Fn 𝐷𝑥𝐷) → ((𝐹𝐹)‘𝑥) = (𝐹‘(𝐹𝑥)))
5847, 57sylan 578 . . 3 ((𝐹𝑅𝑥𝐷) → ((𝐹𝐹)‘𝑥) = (𝐹‘(𝐹𝑥)))
59 fvresi 7172 . . . 4 (𝑥𝐷 → (( I ↾ 𝐷)‘𝑥) = 𝑥)
6059adantl 480 . . 3 ((𝐹𝑅𝑥𝐷) → (( I ↾ 𝐷)‘𝑥) = 𝑥)
6156, 58, 603eqtr4d 2780 . 2 ((𝐹𝑅𝑥𝐷) → ((𝐹𝐹)‘𝑥) = (( I ↾ 𝐷)‘𝑥))
6214, 16, 61eqfnfvd 7034 1 (𝐹𝑅 → (𝐹𝐹) = ( I ↾ 𝐷))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 394  w3a 1085   = wceq 1539  wcel 2104  wne 2938  Vcvv 3472  cdif 3944  wss 3947  ifcif 4527  {csn 4627   cuni 4907   class class class wbr 5147   I cid 5572  dom cdm 5675  ran crn 5676  cres 5677  ccom 5679  suc csuc 6365   Fn wfn 6537  wf 6538  cfv 6542  ωcom 7857  1oc1o 8461  2oc2o 8462  cen 8938  pmTrspcpmtr 19350
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7727
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3375  df-rab 3431  df-v 3474  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 6366  df-on 6367  df-lim 6368  df-suc 6369  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-om 7858  df-1o 8468  df-2o 8469  df-er 8705  df-en 8942  df-dom 8943  df-sdom 8944  df-fin 8945  df-pmtr 19351
This theorem is referenced by:  pmtrff1o  19372  pmtrfcnv  19373  symggen  19379  psgnunilem1  19402  cyc3genpmlem  32580
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