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Theorem pmtrfinv 18082
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 2806 . . . . . . 7 dom (𝐹 ∖ I ) = dom (𝐹 ∖ I )
41, 2, 3pmtrfrn 18079 . . . . . 6 (𝐹𝑅 → ((𝐷 ∈ V ∧ dom (𝐹 ∖ I ) ⊆ 𝐷 ∧ dom (𝐹 ∖ I ) ≈ 2𝑜) ∧ 𝐹 = (𝑇‘dom (𝐹 ∖ I ))))
54simpld 484 . . . . 5 (𝐹𝑅 → (𝐷 ∈ V ∧ dom (𝐹 ∖ I ) ⊆ 𝐷 ∧ dom (𝐹 ∖ I ) ≈ 2𝑜))
61pmtrf 18076 . . . . 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 485 . . . . 5 (𝐹𝑅𝐹 = (𝑇‘dom (𝐹 ∖ I )))
98feq1d 6241 . . . 4 (𝐹𝑅 → (𝐹:𝐷𝐷 ↔ (𝑇‘dom (𝐹 ∖ I )):𝐷𝐷))
107, 9mpbird 248 . . 3 (𝐹𝑅𝐹:𝐷𝐷)
11 fco 6273 . . . 4 ((𝐹:𝐷𝐷𝐹:𝐷𝐷) → (𝐹𝐹):𝐷𝐷)
1211anidms 558 . . 3 (𝐹:𝐷𝐷 → (𝐹𝐹):𝐷𝐷)
13 ffn 6256 . . 3 ((𝐹𝐹):𝐷𝐷 → (𝐹𝐹) Fn 𝐷)
1410, 12, 133syl 18 . 2 (𝐹𝑅 → (𝐹𝐹) Fn 𝐷)
15 fnresi 6219 . . 3 ( I ↾ 𝐷) Fn 𝐷
1615a1i 11 . 2 (𝐹𝑅 → ( I ↾ 𝐷) Fn 𝐷)
171, 2, 3pmtrffv 18080 . . . . . . 7 ((𝐹𝑅𝑥𝐷) → (𝐹𝑥) = if(𝑥 ∈ dom (𝐹 ∖ I ), (dom (𝐹 ∖ I ) ∖ {𝑥}), 𝑥))
18 iftrue 4285 . . . . . . 7 (𝑥 ∈ dom (𝐹 ∖ I ) → if(𝑥 ∈ dom (𝐹 ∖ I ), (dom (𝐹 ∖ I ) ∖ {𝑥}), 𝑥) = (dom (𝐹 ∖ I ) ∖ {𝑥}))
1917, 18sylan9eq 2860 . . . . . 6 (((𝐹𝑅𝑥𝐷) ∧ 𝑥 ∈ dom (𝐹 ∖ I )) → (𝐹𝑥) = (dom (𝐹 ∖ I ) ∖ {𝑥}))
2019fveq2d 6412 . . . . 5 (((𝐹𝑅𝑥𝐷) ∧ 𝑥 ∈ dom (𝐹 ∖ I )) → (𝐹‘(𝐹𝑥)) = (𝐹 (dom (𝐹 ∖ I ) ∖ {𝑥})))
21 simpll 774 . . . . . . 7 (((𝐹𝑅𝑥𝐷) ∧ 𝑥 ∈ dom (𝐹 ∖ I )) → 𝐹𝑅)
225simp2d 1166 . . . . . . . . 9 (𝐹𝑅 → dom (𝐹 ∖ I ) ⊆ 𝐷)
2322ad2antrr 708 . . . . . . . 8 (((𝐹𝑅𝑥𝐷) ∧ 𝑥 ∈ dom (𝐹 ∖ I )) → dom (𝐹 ∖ I ) ⊆ 𝐷)
24 1onn 7956 . . . . . . . . . . . 12 1𝑜 ∈ ω
2524a1i 11 . . . . . . . . . . 11 (((𝐹𝑅𝑥𝐷) ∧ 𝑥 ∈ dom (𝐹 ∖ I )) → 1𝑜 ∈ ω)
265simp3d 1167 . . . . . . . . . . . . 13 (𝐹𝑅 → dom (𝐹 ∖ I ) ≈ 2𝑜)
27 df-2o 7797 . . . . . . . . . . . . 13 2𝑜 = suc 1𝑜
2826, 27syl6breq 4885 . . . . . . . . . . . 12 (𝐹𝑅 → dom (𝐹 ∖ I ) ≈ suc 1𝑜)
2928ad2antrr 708 . . . . . . . . . . 11 (((𝐹𝑅𝑥𝐷) ∧ 𝑥 ∈ dom (𝐹 ∖ I )) → dom (𝐹 ∖ I ) ≈ suc 1𝑜)
30 simpr 473 . . . . . . . . . . 11 (((𝐹𝑅𝑥𝐷) ∧ 𝑥 ∈ dom (𝐹 ∖ I )) → 𝑥 ∈ dom (𝐹 ∖ I ))
31 dif1en 8432 . . . . . . . . . . 11 ((1𝑜 ∈ ω ∧ dom (𝐹 ∖ I ) ≈ suc 1𝑜𝑥 ∈ dom (𝐹 ∖ I )) → (dom (𝐹 ∖ I ) ∖ {𝑥}) ≈ 1𝑜)
3225, 29, 30, 31syl3anc 1483 . . . . . . . . . 10 (((𝐹𝑅𝑥𝐷) ∧ 𝑥 ∈ dom (𝐹 ∖ I )) → (dom (𝐹 ∖ I ) ∖ {𝑥}) ≈ 1𝑜)
33 en1uniel 8264 . . . . . . . . . 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 3781 . . . . . . . 8 (((𝐹𝑅𝑥𝐷) ∧ 𝑥 ∈ dom (𝐹 ∖ I )) → (dom (𝐹 ∖ I ) ∖ {𝑥}) ∈ dom (𝐹 ∖ I ))
3623, 35sseldd 3799 . . . . . . 7 (((𝐹𝑅𝑥𝐷) ∧ 𝑥 ∈ dom (𝐹 ∖ I )) → (dom (𝐹 ∖ I ) ∖ {𝑥}) ∈ 𝐷)
371, 2, 3pmtrffv 18080 . . . . . . 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 575 . . . . . 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 4285 . . . . . . . 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 468 . . . . . . . 8 ((𝐹𝑅𝑥𝐷) → dom (𝐹 ∖ I ) ≈ 2𝑜)
42 en2other2 9115 . . . . . . . . 9 ((𝑥 ∈ dom (𝐹 ∖ I ) ∧ dom (𝐹 ∖ I ) ≈ 2𝑜) → (dom (𝐹 ∖ I ) ∖ { (dom (𝐹 ∖ I ) ∖ {𝑥})}) = 𝑥)
4342ancoms 448 . . . . . . . 8 ((dom (𝐹 ∖ I ) ≈ 2𝑜𝑥 ∈ dom (𝐹 ∖ I )) → (dom (𝐹 ∖ I ) ∖ { (dom (𝐹 ∖ I ) ∖ {𝑥})}) = 𝑥)
4441, 43sylan 571 . . . . . . 7 (((𝐹𝑅𝑥𝐷) ∧ 𝑥 ∈ dom (𝐹 ∖ I )) → (dom (𝐹 ∖ I ) ∖ { (dom (𝐹 ∖ I ) ∖ {𝑥})}) = 𝑥)
4540, 44eqtrd 2840 . . . . . 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 2840 . . . . 5 (((𝐹𝑅𝑥𝐷) ∧ 𝑥 ∈ dom (𝐹 ∖ I )) → (𝐹 (dom (𝐹 ∖ I ) ∖ {𝑥})) = 𝑥)
4720, 46eqtrd 2840 . . . 4 (((𝐹𝑅𝑥𝐷) ∧ 𝑥 ∈ dom (𝐹 ∖ I )) → (𝐹‘(𝐹𝑥)) = 𝑥)
4810ffnd 6257 . . . . . . . 8 (𝐹𝑅𝐹 Fn 𝐷)
49 fnelnfp 6668 . . . . . . . 8 ((𝐹 Fn 𝐷𝑥𝐷) → (𝑥 ∈ dom (𝐹 ∖ I ) ↔ (𝐹𝑥) ≠ 𝑥))
5048, 49sylan 571 . . . . . . 7 ((𝐹𝑅𝑥𝐷) → (𝑥 ∈ dom (𝐹 ∖ I ) ↔ (𝐹𝑥) ≠ 𝑥))
5150necon2bbid 3021 . . . . . 6 ((𝐹𝑅𝑥𝐷) → ((𝐹𝑥) = 𝑥 ↔ ¬ 𝑥 ∈ dom (𝐹 ∖ I )))
5251biimpar 465 . . . . 5 (((𝐹𝑅𝑥𝐷) ∧ ¬ 𝑥 ∈ dom (𝐹 ∖ I )) → (𝐹𝑥) = 𝑥)
53 fveq2 6408 . . . . . 6 ((𝐹𝑥) = 𝑥 → (𝐹‘(𝐹𝑥)) = (𝐹𝑥))
54 id 22 . . . . . 6 ((𝐹𝑥) = 𝑥 → (𝐹𝑥) = 𝑥)
5553, 54eqtrd 2840 . . . . 5 ((𝐹𝑥) = 𝑥 → (𝐹‘(𝐹𝑥)) = 𝑥)
5652, 55syl 17 . . . 4 (((𝐹𝑅𝑥𝐷) ∧ ¬ 𝑥 ∈ dom (𝐹 ∖ I )) → (𝐹‘(𝐹𝑥)) = 𝑥)
5747, 56pm2.61dan 838 . . 3 ((𝐹𝑅𝑥𝐷) → (𝐹‘(𝐹𝑥)) = 𝑥)
58 fvco2 6494 . . . 4 ((𝐹 Fn 𝐷𝑥𝐷) → ((𝐹𝐹)‘𝑥) = (𝐹‘(𝐹𝑥)))
5948, 58sylan 571 . . 3 ((𝐹𝑅𝑥𝐷) → ((𝐹𝐹)‘𝑥) = (𝐹‘(𝐹𝑥)))
60 fvresi 6664 . . . 4 (𝑥𝐷 → (( I ↾ 𝐷)‘𝑥) = 𝑥)
6160adantl 469 . . 3 ((𝐹𝑅𝑥𝐷) → (( I ↾ 𝐷)‘𝑥) = 𝑥)
6257, 59, 613eqtr4d 2850 . 2 ((𝐹𝑅𝑥𝐷) → ((𝐹𝐹)‘𝑥) = (( I ↾ 𝐷)‘𝑥))
6314, 16, 62eqfnfvd 6536 1 (𝐹𝑅 → (𝐹𝐹) = ( I ↾ 𝐷))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 197  wa 384  w3a 1100   = wceq 1637  wcel 2156  wne 2978  Vcvv 3391  cdif 3766  wss 3769  ifcif 4279  {csn 4370   cuni 4630   class class class wbr 4844   I cid 5218  dom cdm 5311  ran crn 5312  cres 5313  ccom 5315  suc csuc 5938   Fn wfn 6096  wf 6097  cfv 6101  ωcom 7295  1𝑜c1o 7789  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-dom 8194  df-sdom 8195  df-fin 8196  df-pmtr 18063
This theorem is referenced by:  pmtrff1o  18084  pmtrfcnv  18085  symggen  18091  psgnunilem1  18114
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