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Theorem dfphi2 4569
Description: Express the phi operator in terms of the Kuratowski set construction functions. (Contributed by SF, 3-Feb-2015.)
Assertion
Ref Expression
dfphi2 Phi A = (((Imagek(( Ins3k ∼ (( Ins3k SkIns2k Sk ) “k 111c) (( Ins2k Ins2k Sk ⊕ ( Ins2k Ins3k SkIns3k SIk SIk Sk )) “k 11111c)) “k 111c) ∩ ( Nn ×k V)) ∪ ( Ik ∩ ( ∼ Nn ×k V))) “k A)

Proof of Theorem dfphi2
Dummy variables x y z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 iftrue 3668 . . . . . . . . 9 (y Nn → if(y Nn , (y +c 1c), y) = (y +c 1c))
21eqeq2d 2364 . . . . . . . 8 (y Nn → (x = if(y Nn , (y +c 1c), y) ↔ x = (y +c 1c)))
3 iba 489 . . . . . . . 8 (y Nn → (x = (y +c 1c) ↔ (x = (y +c 1c) y Nn )))
4 simpr 447 . . . . . . . . . . 11 ((x = y ¬ y Nn ) → ¬ y Nn )
54con2i 112 . . . . . . . . . 10 (y Nn → ¬ (x = y ¬ y Nn ))
6 biorf 394 . . . . . . . . . 10 (¬ (x = y ¬ y Nn ) → ((x = (y +c 1c) y Nn ) ↔ ((x = y ¬ y Nn ) (x = (y +c 1c) y Nn ))))
75, 6syl 15 . . . . . . . . 9 (y Nn → ((x = (y +c 1c) y Nn ) ↔ ((x = y ¬ y Nn ) (x = (y +c 1c) y Nn ))))
8 orcom 376 . . . . . . . . 9 (((x = y ¬ y Nn ) (x = (y +c 1c) y Nn )) ↔ ((x = (y +c 1c) y Nn ) (x = y ¬ y Nn )))
97, 8syl6bb 252 . . . . . . . 8 (y Nn → ((x = (y +c 1c) y Nn ) ↔ ((x = (y +c 1c) y Nn ) (x = y ¬ y Nn ))))
102, 3, 93bitrd 270 . . . . . . 7 (y Nn → (x = if(y Nn , (y +c 1c), y) ↔ ((x = (y +c 1c) y Nn ) (x = y ¬ y Nn ))))
11 iffalse 3669 . . . . . . . . 9 y Nn → if(y Nn , (y +c 1c), y) = y)
1211eqeq2d 2364 . . . . . . . 8 y Nn → (x = if(y Nn , (y +c 1c), y) ↔ x = y))
13 iba 489 . . . . . . . 8 y Nn → (x = y ↔ (x = y ¬ y Nn )))
14 simpr 447 . . . . . . . . . 10 ((x = (y +c 1c) y Nn ) → y Nn )
1514con3i 127 . . . . . . . . 9 y Nn → ¬ (x = (y +c 1c) y Nn ))
16 biorf 394 . . . . . . . . 9 (¬ (x = (y +c 1c) y Nn ) → ((x = y ¬ y Nn ) ↔ ((x = (y +c 1c) y Nn ) (x = y ¬ y Nn ))))
1715, 16syl 15 . . . . . . . 8 y Nn → ((x = y ¬ y Nn ) ↔ ((x = (y +c 1c) y Nn ) (x = y ¬ y Nn ))))
1812, 13, 173bitrd 270 . . . . . . 7 y Nn → (x = if(y Nn , (y +c 1c), y) ↔ ((x = (y +c 1c) y Nn ) (x = y ¬ y Nn ))))
1910, 18pm2.61i 156 . . . . . 6 (x = if(y Nn , (y +c 1c), y) ↔ ((x = (y +c 1c) y Nn ) (x = y ¬ y Nn )))
20 equcom 1680 . . . . . . . 8 (y = xx = y)
21 vex 2862 . . . . . . . . 9 y V
2221elcompl 3225 . . . . . . . 8 (y Nn ↔ ¬ y Nn )
2320, 22anbi12i 678 . . . . . . 7 ((y = x y Nn ) ↔ (x = y ¬ y Nn ))
2423orbi2i 505 . . . . . 6 (((x = (y +c 1c) y Nn ) (y = x y Nn )) ↔ ((x = (y +c 1c) y Nn ) (x = y ¬ y Nn )))
2519, 24bitr4i 243 . . . . 5 (x = if(y Nn , (y +c 1c), y) ↔ ((x = (y +c 1c) y Nn ) (y = x y Nn )))
26 elun 3220 . . . . . 6 (⟪y, x ((Imagek(( Ins3k ∼ (( Ins3k SkIns2k Sk ) “k 111c) (( Ins2k Ins2k Sk ⊕ ( Ins2k Ins3k SkIns3k SIk SIk Sk )) “k 11111c)) “k 111c) ∩ ( Nn ×k V)) ∪ ( Ik ∩ ( ∼ Nn ×k V))) ↔ (⟪y, x (Imagek(( Ins3k ∼ (( Ins3k SkIns2k Sk ) “k 111c) (( Ins2k Ins2k Sk ⊕ ( Ins2k Ins3k SkIns3k SIk SIk Sk )) “k 11111c)) “k 111c) ∩ ( Nn ×k V)) y, x ( Ik ∩ ( ∼ Nn ×k V))))
27 elin 3219 . . . . . . . 8 (⟪y, x (Imagek(( Ins3k ∼ (( Ins3k SkIns2k Sk ) “k 111c) (( Ins2k Ins2k Sk ⊕ ( Ins2k Ins3k SkIns3k SIk SIk Sk )) “k 11111c)) “k 111c) ∩ ( Nn ×k V)) ↔ (⟪y, x Imagek(( Ins3k ∼ (( Ins3k SkIns2k Sk ) “k 111c) (( Ins2k Ins2k Sk ⊕ ( Ins2k Ins3k SkIns3k SIk SIk Sk )) “k 11111c)) “k 111c) y, x ( Nn ×k V)))
28 vex 2862 . . . . . . . . . . 11 x V
2921, 28opkelimagek 4272 . . . . . . . . . 10 (⟪y, x Imagek(( Ins3k ∼ (( Ins3k SkIns2k Sk ) “k 111c) (( Ins2k Ins2k Sk ⊕ ( Ins2k Ins3k SkIns3k SIk SIk Sk )) “k 11111c)) “k 111c) ↔ x = ((( Ins3k ∼ (( Ins3k SkIns2k Sk ) “k 111c) (( Ins2k Ins2k Sk ⊕ ( Ins2k Ins3k SkIns3k SIk SIk Sk )) “k 11111c)) “k 111c) “k y))
30 dfaddc2 4381 . . . . . . . . . . 11 (y +c 1c) = ((( Ins3k ∼ (( Ins3k SkIns2k Sk ) “k 111c) (( Ins2k Ins2k Sk ⊕ ( Ins2k Ins3k SkIns3k SIk SIk Sk )) “k 11111c)) “k 111c) “k y)
3130eqeq2i 2363 . . . . . . . . . 10 (x = (y +c 1c) ↔ x = ((( Ins3k ∼ (( Ins3k SkIns2k Sk ) “k 111c) (( Ins2k Ins2k Sk ⊕ ( Ins2k Ins3k SkIns3k SIk SIk Sk )) “k 11111c)) “k 111c) “k y))
3229, 31bitr4i 243 . . . . . . . . 9 (⟪y, x Imagek(( Ins3k ∼ (( Ins3k SkIns2k Sk ) “k 111c) (( Ins2k Ins2k Sk ⊕ ( Ins2k Ins3k SkIns3k SIk SIk Sk )) “k 11111c)) “k 111c) ↔ x = (y +c 1c))
3321, 28opkelxpk 4248 . . . . . . . . . 10 (⟪y, x ( Nn ×k V) ↔ (y Nn x V))
3428, 33mpbiran2 885 . . . . . . . . 9 (⟪y, x ( Nn ×k V) ↔ y Nn )
3532, 34anbi12i 678 . . . . . . . 8 ((⟪y, x Imagek(( Ins3k ∼ (( Ins3k SkIns2k Sk ) “k 111c) (( Ins2k Ins2k Sk ⊕ ( Ins2k Ins3k SkIns3k SIk SIk Sk )) “k 11111c)) “k 111c) y, x ( Nn ×k V)) ↔ (x = (y +c 1c) y Nn ))
3627, 35bitri 240 . . . . . . 7 (⟪y, x (Imagek(( Ins3k ∼ (( Ins3k SkIns2k Sk ) “k 111c) (( Ins2k Ins2k Sk ⊕ ( Ins2k Ins3k SkIns3k SIk SIk Sk )) “k 11111c)) “k 111c) ∩ ( Nn ×k V)) ↔ (x = (y +c 1c) y Nn ))
37 elin 3219 . . . . . . . 8 (⟪y, x ( Ik ∩ ( ∼ Nn ×k V)) ↔ (⟪y, x Ik y, x ( ∼ Nn ×k V)))
38 opkelidkg 4274 . . . . . . . . . 10 ((y V x V) → (⟪y, x Iky = x))
3921, 28, 38mp2an 653 . . . . . . . . 9 (⟪y, x Iky = x)
4021, 28opkelxpk 4248 . . . . . . . . . 10 (⟪y, x ( ∼ Nn ×k V) ↔ (y Nn x V))
4128, 40mpbiran2 885 . . . . . . . . 9 (⟪y, x ( ∼ Nn ×k V) ↔ y Nn )
4239, 41anbi12i 678 . . . . . . . 8 ((⟪y, x Ik y, x ( ∼ Nn ×k V)) ↔ (y = x y Nn ))
4337, 42bitri 240 . . . . . . 7 (⟪y, x ( Ik ∩ ( ∼ Nn ×k V)) ↔ (y = x y Nn ))
4436, 43orbi12i 507 . . . . . 6 ((⟪y, x (Imagek(( Ins3k ∼ (( Ins3k SkIns2k Sk ) “k 111c) (( Ins2k Ins2k Sk ⊕ ( Ins2k Ins3k SkIns3k SIk SIk Sk )) “k 11111c)) “k 111c) ∩ ( Nn ×k V)) y, x ( Ik ∩ ( ∼ Nn ×k V))) ↔ ((x = (y +c 1c) y Nn ) (y = x y Nn )))
4526, 44bitri 240 . . . . 5 (⟪y, x ((Imagek(( Ins3k ∼ (( Ins3k SkIns2k Sk ) “k 111c) (( Ins2k Ins2k Sk ⊕ ( Ins2k Ins3k SkIns3k SIk SIk Sk )) “k 11111c)) “k 111c) ∩ ( Nn ×k V)) ∪ ( Ik ∩ ( ∼ Nn ×k V))) ↔ ((x = (y +c 1c) y Nn ) (y = x y Nn )))
4625, 45bitr4i 243 . . . 4 (x = if(y Nn , (y +c 1c), y) ↔ ⟪y, x ((Imagek(( Ins3k ∼ (( Ins3k SkIns2k Sk ) “k 111c) (( Ins2k Ins2k Sk ⊕ ( Ins2k Ins3k SkIns3k SIk SIk Sk )) “k 11111c)) “k 111c) ∩ ( Nn ×k V)) ∪ ( Ik ∩ ( ∼ Nn ×k V))))
4746rexbii 2639 . . 3 (y A x = if(y Nn , (y +c 1c), y) ↔ y Ay, x ((Imagek(( Ins3k ∼ (( Ins3k SkIns2k Sk ) “k 111c) (( Ins2k Ins2k Sk ⊕ ( Ins2k Ins3k SkIns3k SIk SIk Sk )) “k 11111c)) “k 111c) ∩ ( Nn ×k V)) ∪ ( Ik ∩ ( ∼ Nn ×k V))))
48 eqeq1 2359 . . . . 5 (z = x → (z = if(y Nn , (y +c 1c), y) ↔ x = if(y Nn , (y +c 1c), y)))
4948rexbidv 2635 . . . 4 (z = x → (y A z = if(y Nn , (y +c 1c), y) ↔ y A x = if(y Nn , (y +c 1c), y)))
50 df-phi 4565 . . . 4 Phi A = {z y A z = if(y Nn , (y +c 1c), y)}
5128, 49, 50elab2 2988 . . 3 (x Phi Ay A x = if(y Nn , (y +c 1c), y))
5228elimak 4259 . . 3 (x (((Imagek(( Ins3k ∼ (( Ins3k SkIns2k Sk ) “k 111c) (( Ins2k Ins2k Sk ⊕ ( Ins2k Ins3k SkIns3k SIk SIk Sk )) “k 11111c)) “k 111c) ∩ ( Nn ×k V)) ∪ ( Ik ∩ ( ∼ Nn ×k V))) “k A) ↔ y Ay, x ((Imagek(( Ins3k ∼ (( Ins3k SkIns2k Sk ) “k 111c) (( Ins2k Ins2k Sk ⊕ ( Ins2k Ins3k SkIns3k SIk SIk Sk )) “k 11111c)) “k 111c) ∩ ( Nn ×k V)) ∪ ( Ik ∩ ( ∼ Nn ×k V))))
5347, 51, 523bitr4i 268 . 2 (x Phi Ax (((Imagek(( Ins3k ∼ (( Ins3k SkIns2k Sk ) “k 111c) (( Ins2k Ins2k Sk ⊕ ( Ins2k Ins3k SkIns3k SIk SIk Sk )) “k 11111c)) “k 111c) ∩ ( Nn ×k V)) ∪ ( Ik ∩ ( ∼ Nn ×k V))) “k A))
5453eqriv 2350 1 Phi A = (((Imagek(( Ins3k ∼ (( Ins3k SkIns2k Sk ) “k 111c) (( Ins2k Ins2k Sk ⊕ ( Ins2k Ins3k SkIns3k SIk SIk Sk )) “k 11111c)) “k 111c) ∩ ( Nn ×k V)) ∪ ( Ik ∩ ( ∼ Nn ×k V))) “k A)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 176   wo 357   wa 358   = wceq 1642   wcel 1710  wrex 2615  Vcvv 2859  ccompl 3205   cdif 3206  cun 3207  cin 3208  csymdif 3209   ifcif 3662  copk 4057  1cc1c 4134  1cpw1 4135   ×k cxpk 4174   Ins2k cins2k 4176   Ins3k cins3k 4177  k cimak 4179   SIk csik 4181  Imagekcimagek 4182   Sk cssetk 4183   Ik cidk 4184   Nn cnnc 4373   +c cplc 4375   Phi cphi 4562
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334  ax-nin 4078  ax-sn 4087
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-ne 2518  df-ral 2619  df-rex 2620  df-v 2861  df-nin 3211  df-compl 3212  df-in 3213  df-un 3214  df-dif 3215  df-symdif 3216  df-ss 3259  df-nul 3551  df-if 3663  df-pw 3724  df-sn 3741  df-pr 3742  df-opk 4058  df-1c 4136  df-pw1 4137  df-xpk 4185  df-cnvk 4186  df-ins2k 4187  df-ins3k 4188  df-imak 4189  df-cok 4190  df-sik 4192  df-ssetk 4193  df-imagek 4194  df-idk 4195  df-addc 4378  df-phi 4565
This theorem is referenced by:  phieq  4570  phiexg  4571  dfop2lem1  4573  dfop2  4575  dfproj12  4576  setconslem1  4731  setconslem2  4732  dfswap2  4741
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