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Theorem pmtrfinv 18231
 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 2825 . . . . . . 7 dom (𝐹 ∖ I ) = dom (𝐹 ∖ I )
41, 2, 3pmtrfrn 18228 . . . . . 6 (𝐹𝑅 → ((𝐷 ∈ V ∧ dom (𝐹 ∖ I ) ⊆ 𝐷 ∧ dom (𝐹 ∖ I ) ≈ 2o) ∧ 𝐹 = (𝑇‘dom (𝐹 ∖ I ))))
54simpld 490 . . . . 5 (𝐹𝑅 → (𝐷 ∈ V ∧ dom (𝐹 ∖ I ) ⊆ 𝐷 ∧ dom (𝐹 ∖ I ) ≈ 2o))
61pmtrf 18225 . . . . 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 491 . . . . 5 (𝐹𝑅𝐹 = (𝑇‘dom (𝐹 ∖ I )))
98feq1d 6263 . . . 4 (𝐹𝑅 → (𝐹:𝐷𝐷 ↔ (𝑇‘dom (𝐹 ∖ I )):𝐷𝐷))
107, 9mpbird 249 . . 3 (𝐹𝑅𝐹:𝐷𝐷)
11 fco 6295 . . . 4 ((𝐹:𝐷𝐷𝐹:𝐷𝐷) → (𝐹𝐹):𝐷𝐷)
1211anidms 562 . . 3 (𝐹:𝐷𝐷 → (𝐹𝐹):𝐷𝐷)
13 ffn 6278 . . 3 ((𝐹𝐹):𝐷𝐷 → (𝐹𝐹) Fn 𝐷)
1410, 12, 133syl 18 . 2 (𝐹𝑅 → (𝐹𝐹) Fn 𝐷)
15 fnresi 6241 . . 3 ( I ↾ 𝐷) Fn 𝐷
1615a1i 11 . 2 (𝐹𝑅 → ( I ↾ 𝐷) Fn 𝐷)
171, 2, 3pmtrffv 18229 . . . . . . 7 ((𝐹𝑅𝑥𝐷) → (𝐹𝑥) = if(𝑥 ∈ dom (𝐹 ∖ I ), (dom (𝐹 ∖ I ) ∖ {𝑥}), 𝑥))
18 iftrue 4312 . . . . . . 7 (𝑥 ∈ dom (𝐹 ∖ I ) → if(𝑥 ∈ dom (𝐹 ∖ I ), (dom (𝐹 ∖ I ) ∖ {𝑥}), 𝑥) = (dom (𝐹 ∖ I ) ∖ {𝑥}))
1917, 18sylan9eq 2881 . . . . . 6 (((𝐹𝑅𝑥𝐷) ∧ 𝑥 ∈ dom (𝐹 ∖ I )) → (𝐹𝑥) = (dom (𝐹 ∖ I ) ∖ {𝑥}))
2019fveq2d 6437 . . . . 5 (((𝐹𝑅𝑥𝐷) ∧ 𝑥 ∈ dom (𝐹 ∖ I )) → (𝐹‘(𝐹𝑥)) = (𝐹 (dom (𝐹 ∖ I ) ∖ {𝑥})))
21 simpll 783 . . . . . . 7 (((𝐹𝑅𝑥𝐷) ∧ 𝑥 ∈ dom (𝐹 ∖ I )) → 𝐹𝑅)
225simp2d 1177 . . . . . . . . 9 (𝐹𝑅 → dom (𝐹 ∖ I ) ⊆ 𝐷)
2322ad2antrr 717 . . . . . . . 8 (((𝐹𝑅𝑥𝐷) ∧ 𝑥 ∈ dom (𝐹 ∖ I )) → dom (𝐹 ∖ I ) ⊆ 𝐷)
24 1onn 7986 . . . . . . . . . . . 12 1o ∈ ω
2524a1i 11 . . . . . . . . . . 11 (((𝐹𝑅𝑥𝐷) ∧ 𝑥 ∈ dom (𝐹 ∖ I )) → 1o ∈ ω)
265simp3d 1178 . . . . . . . . . . . . 13 (𝐹𝑅 → dom (𝐹 ∖ I ) ≈ 2o)
27 df-2o 7827 . . . . . . . . . . . . 13 2o = suc 1o
2826, 27syl6breq 4914 . . . . . . . . . . . 12 (𝐹𝑅 → dom (𝐹 ∖ I ) ≈ suc 1o)
2928ad2antrr 717 . . . . . . . . . . 11 (((𝐹𝑅𝑥𝐷) ∧ 𝑥 ∈ dom (𝐹 ∖ I )) → dom (𝐹 ∖ I ) ≈ suc 1o)
30 simpr 479 . . . . . . . . . . 11 (((𝐹𝑅𝑥𝐷) ∧ 𝑥 ∈ dom (𝐹 ∖ I )) → 𝑥 ∈ dom (𝐹 ∖ I ))
31 dif1en 8462 . . . . . . . . . . 11 ((1o ∈ ω ∧ dom (𝐹 ∖ I ) ≈ suc 1o𝑥 ∈ dom (𝐹 ∖ I )) → (dom (𝐹 ∖ I ) ∖ {𝑥}) ≈ 1o)
3225, 29, 30, 31syl3anc 1494 . . . . . . . . . 10 (((𝐹𝑅𝑥𝐷) ∧ 𝑥 ∈ dom (𝐹 ∖ I )) → (dom (𝐹 ∖ I ) ∖ {𝑥}) ≈ 1o)
33 en1uniel 8294 . . . . . . . . . 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 3810 . . . . . . . 8 (((𝐹𝑅𝑥𝐷) ∧ 𝑥 ∈ dom (𝐹 ∖ I )) → (dom (𝐹 ∖ I ) ∖ {𝑥}) ∈ dom (𝐹 ∖ I ))
3623, 35sseldd 3828 . . . . . . 7 (((𝐹𝑅𝑥𝐷) ∧ 𝑥 ∈ dom (𝐹 ∖ I )) → (dom (𝐹 ∖ I ) ∖ {𝑥}) ∈ 𝐷)
371, 2, 3pmtrffv 18229 . . . . . . 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 579 . . . . . 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 4312 . . . . . . . 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 474 . . . . . . . 8 ((𝐹𝑅𝑥𝐷) → dom (𝐹 ∖ I ) ≈ 2o)
42 en2other2 9145 . . . . . . . . 9 ((𝑥 ∈ dom (𝐹 ∖ I ) ∧ dom (𝐹 ∖ I ) ≈ 2o) → (dom (𝐹 ∖ I ) ∖ { (dom (𝐹 ∖ I ) ∖ {𝑥})}) = 𝑥)
4342ancoms 452 . . . . . . . 8 ((dom (𝐹 ∖ I ) ≈ 2o𝑥 ∈ dom (𝐹 ∖ I )) → (dom (𝐹 ∖ I ) ∖ { (dom (𝐹 ∖ I ) ∖ {𝑥})}) = 𝑥)
4441, 43sylan 575 . . . . . . 7 (((𝐹𝑅𝑥𝐷) ∧ 𝑥 ∈ dom (𝐹 ∖ I )) → (dom (𝐹 ∖ I ) ∖ { (dom (𝐹 ∖ I ) ∖ {𝑥})}) = 𝑥)
4540, 44eqtrd 2861 . . . . . 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 2861 . . . . 5 (((𝐹𝑅𝑥𝐷) ∧ 𝑥 ∈ dom (𝐹 ∖ I )) → (𝐹 (dom (𝐹 ∖ I ) ∖ {𝑥})) = 𝑥)
4720, 46eqtrd 2861 . . . 4 (((𝐹𝑅𝑥𝐷) ∧ 𝑥 ∈ dom (𝐹 ∖ I )) → (𝐹‘(𝐹𝑥)) = 𝑥)
4810ffnd 6279 . . . . . . . 8 (𝐹𝑅𝐹 Fn 𝐷)
49 fnelnfp 6695 . . . . . . . 8 ((𝐹 Fn 𝐷𝑥𝐷) → (𝑥 ∈ dom (𝐹 ∖ I ) ↔ (𝐹𝑥) ≠ 𝑥))
5048, 49sylan 575 . . . . . . 7 ((𝐹𝑅𝑥𝐷) → (𝑥 ∈ dom (𝐹 ∖ I ) ↔ (𝐹𝑥) ≠ 𝑥))
5150necon2bbid 3042 . . . . . 6 ((𝐹𝑅𝑥𝐷) → ((𝐹𝑥) = 𝑥 ↔ ¬ 𝑥 ∈ dom (𝐹 ∖ I )))
5251biimpar 471 . . . . 5 (((𝐹𝑅𝑥𝐷) ∧ ¬ 𝑥 ∈ dom (𝐹 ∖ I )) → (𝐹𝑥) = 𝑥)
53 fveq2 6433 . . . . . 6 ((𝐹𝑥) = 𝑥 → (𝐹‘(𝐹𝑥)) = (𝐹𝑥))
54 id 22 . . . . . 6 ((𝐹𝑥) = 𝑥 → (𝐹𝑥) = 𝑥)
5553, 54eqtrd 2861 . . . . 5 ((𝐹𝑥) = 𝑥 → (𝐹‘(𝐹𝑥)) = 𝑥)
5652, 55syl 17 . . . 4 (((𝐹𝑅𝑥𝐷) ∧ ¬ 𝑥 ∈ dom (𝐹 ∖ I )) → (𝐹‘(𝐹𝑥)) = 𝑥)
5747, 56pm2.61dan 847 . . 3 ((𝐹𝑅𝑥𝐷) → (𝐹‘(𝐹𝑥)) = 𝑥)
58 fvco2 6520 . . . 4 ((𝐹 Fn 𝐷𝑥𝐷) → ((𝐹𝐹)‘𝑥) = (𝐹‘(𝐹𝑥)))
5948, 58sylan 575 . . 3 ((𝐹𝑅𝑥𝐷) → ((𝐹𝐹)‘𝑥) = (𝐹‘(𝐹𝑥)))
60 fvresi 6691 . . . 4 (𝑥𝐷 → (( I ↾ 𝐷)‘𝑥) = 𝑥)
6160adantl 475 . . 3 ((𝐹𝑅𝑥𝐷) → (( I ↾ 𝐷)‘𝑥) = 𝑥)
6257, 59, 613eqtr4d 2871 . 2 ((𝐹𝑅𝑥𝐷) → ((𝐹𝐹)‘𝑥) = (( I ↾ 𝐷)‘𝑥))
6314, 16, 62eqfnfvd 6563 1 (𝐹𝑅 → (𝐹𝐹) = ( I ↾ 𝐷))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 198   ∧ wa 386   ∧ w3a 1111   = wceq 1656   ∈ wcel 2164   ≠ wne 2999  Vcvv 3414   ∖ cdif 3795   ⊆ wss 3798  ifcif 4306  {csn 4397  ∪ cuni 4658   class class class wbr 4873   I cid 5249  dom cdm 5342  ran crn 5343   ↾ cres 5344   ∘ ccom 5346  suc csuc 5965   Fn wfn 6118  ⟶wf 6119  ‘cfv 6123  ωcom 7326  1oc1o 7819  2oc2o 7820   ≈ cen 8219  pmTrspcpmtr 18211 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-8 2166  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803  ax-rep 4994  ax-sep 5005  ax-nul 5013  ax-pow 5065  ax-pr 5127  ax-un 7209 This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-3or 1112  df-3an 1113  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ne 3000  df-ral 3122  df-rex 3123  df-reu 3124  df-rab 3126  df-v 3416  df-sbc 3663  df-csb 3758  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-pss 3814  df-nul 4145  df-if 4307  df-pw 4380  df-sn 4398  df-pr 4400  df-tp 4402  df-op 4404  df-uni 4659  df-iun 4742  df-br 4874  df-opab 4936  df-mpt 4953  df-tr 4976  df-id 5250  df-eprel 5255  df-po 5263  df-so 5264  df-fr 5301  df-we 5303  df-xp 5348  df-rel 5349  df-cnv 5350  df-co 5351  df-dm 5352  df-rn 5353  df-res 5354  df-ima 5355  df-ord 5966  df-on 5967  df-lim 5968  df-suc 5969  df-iota 6086  df-fun 6125  df-fn 6126  df-f 6127  df-f1 6128  df-fo 6129  df-f1o 6130  df-fv 6131  df-om 7327  df-1o 7826  df-2o 7827  df-er 8009  df-en 8223  df-dom 8224  df-sdom 8225  df-fin 8226  df-pmtr 18212 This theorem is referenced by:  pmtrff1o  18233  pmtrfcnv  18234  symggen  18240  psgnunilem1  18263
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