ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  sbcfung GIF version

Theorem sbcfung 5117
Description: Distribute proper substitution through the function predicate. (Contributed by Alexander van der Vekens, 23-Jul-2017.)
Assertion
Ref Expression
sbcfung (𝐴𝑉 → ([𝐴 / 𝑥]Fun 𝐹 ↔ Fun 𝐴 / 𝑥𝐹))

Proof of Theorem sbcfung
Dummy variables 𝑤 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 sbcan 2923 . . 3 ([𝐴 / 𝑥](Rel 𝐹 ∧ ∀𝑤𝑦𝑧((𝑤𝐹𝑦𝑤𝐹𝑧) → 𝑦 = 𝑧)) ↔ ([𝐴 / 𝑥]Rel 𝐹[𝐴 / 𝑥]𝑤𝑦𝑧((𝑤𝐹𝑦𝑤𝐹𝑧) → 𝑦 = 𝑧)))
2 sbcrel 4595 . . . 4 (𝐴𝑉 → ([𝐴 / 𝑥]Rel 𝐹 ↔ Rel 𝐴 / 𝑥𝐹))
3 sbcal 2932 . . . . 5 ([𝐴 / 𝑥]𝑤𝑦𝑧((𝑤𝐹𝑦𝑤𝐹𝑧) → 𝑦 = 𝑧) ↔ ∀𝑤[𝐴 / 𝑥]𝑦𝑧((𝑤𝐹𝑦𝑤𝐹𝑧) → 𝑦 = 𝑧))
4 sbcal 2932 . . . . . . 7 ([𝐴 / 𝑥]𝑦𝑧((𝑤𝐹𝑦𝑤𝐹𝑧) → 𝑦 = 𝑧) ↔ ∀𝑦[𝐴 / 𝑥]𝑧((𝑤𝐹𝑦𝑤𝐹𝑧) → 𝑦 = 𝑧))
5 sbcal 2932 . . . . . . . . 9 ([𝐴 / 𝑥]𝑧((𝑤𝐹𝑦𝑤𝐹𝑧) → 𝑦 = 𝑧) ↔ ∀𝑧[𝐴 / 𝑥]((𝑤𝐹𝑦𝑤𝐹𝑧) → 𝑦 = 𝑧))
6 sbcimg 2922 . . . . . . . . . . 11 (𝐴𝑉 → ([𝐴 / 𝑥]((𝑤𝐹𝑦𝑤𝐹𝑧) → 𝑦 = 𝑧) ↔ ([𝐴 / 𝑥](𝑤𝐹𝑦𝑤𝐹𝑧) → [𝐴 / 𝑥]𝑦 = 𝑧)))
7 sbcan 2923 . . . . . . . . . . . . 13 ([𝐴 / 𝑥](𝑤𝐹𝑦𝑤𝐹𝑧) ↔ ([𝐴 / 𝑥]𝑤𝐹𝑦[𝐴 / 𝑥]𝑤𝐹𝑧))
8 sbcbrg 3952 . . . . . . . . . . . . . . 15 (𝐴𝑉 → ([𝐴 / 𝑥]𝑤𝐹𝑦𝐴 / 𝑥𝑤𝐴 / 𝑥𝐹𝐴 / 𝑥𝑦))
9 csbconstg 2987 . . . . . . . . . . . . . . . 16 (𝐴𝑉𝐴 / 𝑥𝑤 = 𝑤)
10 csbconstg 2987 . . . . . . . . . . . . . . . 16 (𝐴𝑉𝐴 / 𝑥𝑦 = 𝑦)
119, 10breq12d 3912 . . . . . . . . . . . . . . 15 (𝐴𝑉 → (𝐴 / 𝑥𝑤𝐴 / 𝑥𝐹𝐴 / 𝑥𝑦𝑤𝐴 / 𝑥𝐹𝑦))
128, 11bitrd 187 . . . . . . . . . . . . . 14 (𝐴𝑉 → ([𝐴 / 𝑥]𝑤𝐹𝑦𝑤𝐴 / 𝑥𝐹𝑦))
13 sbcbrg 3952 . . . . . . . . . . . . . . 15 (𝐴𝑉 → ([𝐴 / 𝑥]𝑤𝐹𝑧𝐴 / 𝑥𝑤𝐴 / 𝑥𝐹𝐴 / 𝑥𝑧))
14 csbconstg 2987 . . . . . . . . . . . . . . . 16 (𝐴𝑉𝐴 / 𝑥𝑧 = 𝑧)
159, 14breq12d 3912 . . . . . . . . . . . . . . 15 (𝐴𝑉 → (𝐴 / 𝑥𝑤𝐴 / 𝑥𝐹𝐴 / 𝑥𝑧𝑤𝐴 / 𝑥𝐹𝑧))
1613, 15bitrd 187 . . . . . . . . . . . . . 14 (𝐴𝑉 → ([𝐴 / 𝑥]𝑤𝐹𝑧𝑤𝐴 / 𝑥𝐹𝑧))
1712, 16anbi12d 464 . . . . . . . . . . . . 13 (𝐴𝑉 → (([𝐴 / 𝑥]𝑤𝐹𝑦[𝐴 / 𝑥]𝑤𝐹𝑧) ↔ (𝑤𝐴 / 𝑥𝐹𝑦𝑤𝐴 / 𝑥𝐹𝑧)))
187, 17syl5bb 191 . . . . . . . . . . . 12 (𝐴𝑉 → ([𝐴 / 𝑥](𝑤𝐹𝑦𝑤𝐹𝑧) ↔ (𝑤𝐴 / 𝑥𝐹𝑦𝑤𝐴 / 𝑥𝐹𝑧)))
19 sbcg 2950 . . . . . . . . . . . 12 (𝐴𝑉 → ([𝐴 / 𝑥]𝑦 = 𝑧𝑦 = 𝑧))
2018, 19imbi12d 233 . . . . . . . . . . 11 (𝐴𝑉 → (([𝐴 / 𝑥](𝑤𝐹𝑦𝑤𝐹𝑧) → [𝐴 / 𝑥]𝑦 = 𝑧) ↔ ((𝑤𝐴 / 𝑥𝐹𝑦𝑤𝐴 / 𝑥𝐹𝑧) → 𝑦 = 𝑧)))
216, 20bitrd 187 . . . . . . . . . 10 (𝐴𝑉 → ([𝐴 / 𝑥]((𝑤𝐹𝑦𝑤𝐹𝑧) → 𝑦 = 𝑧) ↔ ((𝑤𝐴 / 𝑥𝐹𝑦𝑤𝐴 / 𝑥𝐹𝑧) → 𝑦 = 𝑧)))
2221albidv 1780 . . . . . . . . 9 (𝐴𝑉 → (∀𝑧[𝐴 / 𝑥]((𝑤𝐹𝑦𝑤𝐹𝑧) → 𝑦 = 𝑧) ↔ ∀𝑧((𝑤𝐴 / 𝑥𝐹𝑦𝑤𝐴 / 𝑥𝐹𝑧) → 𝑦 = 𝑧)))
235, 22syl5bb 191 . . . . . . . 8 (𝐴𝑉 → ([𝐴 / 𝑥]𝑧((𝑤𝐹𝑦𝑤𝐹𝑧) → 𝑦 = 𝑧) ↔ ∀𝑧((𝑤𝐴 / 𝑥𝐹𝑦𝑤𝐴 / 𝑥𝐹𝑧) → 𝑦 = 𝑧)))
2423albidv 1780 . . . . . . 7 (𝐴𝑉 → (∀𝑦[𝐴 / 𝑥]𝑧((𝑤𝐹𝑦𝑤𝐹𝑧) → 𝑦 = 𝑧) ↔ ∀𝑦𝑧((𝑤𝐴 / 𝑥𝐹𝑦𝑤𝐴 / 𝑥𝐹𝑧) → 𝑦 = 𝑧)))
254, 24syl5bb 191 . . . . . 6 (𝐴𝑉 → ([𝐴 / 𝑥]𝑦𝑧((𝑤𝐹𝑦𝑤𝐹𝑧) → 𝑦 = 𝑧) ↔ ∀𝑦𝑧((𝑤𝐴 / 𝑥𝐹𝑦𝑤𝐴 / 𝑥𝐹𝑧) → 𝑦 = 𝑧)))
2625albidv 1780 . . . . 5 (𝐴𝑉 → (∀𝑤[𝐴 / 𝑥]𝑦𝑧((𝑤𝐹𝑦𝑤𝐹𝑧) → 𝑦 = 𝑧) ↔ ∀𝑤𝑦𝑧((𝑤𝐴 / 𝑥𝐹𝑦𝑤𝐴 / 𝑥𝐹𝑧) → 𝑦 = 𝑧)))
273, 26syl5bb 191 . . . 4 (𝐴𝑉 → ([𝐴 / 𝑥]𝑤𝑦𝑧((𝑤𝐹𝑦𝑤𝐹𝑧) → 𝑦 = 𝑧) ↔ ∀𝑤𝑦𝑧((𝑤𝐴 / 𝑥𝐹𝑦𝑤𝐴 / 𝑥𝐹𝑧) → 𝑦 = 𝑧)))
282, 27anbi12d 464 . . 3 (𝐴𝑉 → (([𝐴 / 𝑥]Rel 𝐹[𝐴 / 𝑥]𝑤𝑦𝑧((𝑤𝐹𝑦𝑤𝐹𝑧) → 𝑦 = 𝑧)) ↔ (Rel 𝐴 / 𝑥𝐹 ∧ ∀𝑤𝑦𝑧((𝑤𝐴 / 𝑥𝐹𝑦𝑤𝐴 / 𝑥𝐹𝑧) → 𝑦 = 𝑧))))
291, 28syl5bb 191 . 2 (𝐴𝑉 → ([𝐴 / 𝑥](Rel 𝐹 ∧ ∀𝑤𝑦𝑧((𝑤𝐹𝑦𝑤𝐹𝑧) → 𝑦 = 𝑧)) ↔ (Rel 𝐴 / 𝑥𝐹 ∧ ∀𝑤𝑦𝑧((𝑤𝐴 / 𝑥𝐹𝑦𝑤𝐴 / 𝑥𝐹𝑧) → 𝑦 = 𝑧))))
30 dffun2 5103 . . 3 (Fun 𝐹 ↔ (Rel 𝐹 ∧ ∀𝑤𝑦𝑧((𝑤𝐹𝑦𝑤𝐹𝑧) → 𝑦 = 𝑧)))
3130sbcbii 2940 . 2 ([𝐴 / 𝑥]Fun 𝐹[𝐴 / 𝑥](Rel 𝐹 ∧ ∀𝑤𝑦𝑧((𝑤𝐹𝑦𝑤𝐹𝑧) → 𝑦 = 𝑧)))
32 dffun2 5103 . 2 (Fun 𝐴 / 𝑥𝐹 ↔ (Rel 𝐴 / 𝑥𝐹 ∧ ∀𝑤𝑦𝑧((𝑤𝐴 / 𝑥𝐹𝑦𝑤𝐴 / 𝑥𝐹𝑧) → 𝑦 = 𝑧)))
3329, 31, 323bitr4g 222 1 (𝐴𝑉 → ([𝐴 / 𝑥]Fun 𝐹 ↔ Fun 𝐴 / 𝑥𝐹))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wb 104  wal 1314  wcel 1465  [wsbc 2882  csb 2975   class class class wbr 3899  Rel wrel 4514  Fun wfun 5087
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016  ax-pow 4068  ax-pr 4101
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ral 2398  df-v 2662  df-sbc 2883  df-csb 2976  df-un 3045  df-in 3047  df-ss 3054  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-br 3900  df-opab 3960  df-id 4185  df-rel 4516  df-cnv 4517  df-co 4518  df-fun 5095
This theorem is referenced by:  sbcfng  5240
  Copyright terms: Public domain W3C validator