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

Theorem sbcfung 4953
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 2828 . . 3 ([𝐴 / 𝑥](Rel 𝐹 ∧ ∀𝑤𝑦𝑧((𝑤𝐹𝑦𝑤𝐹𝑧) → 𝑦 = 𝑧)) ↔ ([𝐴 / 𝑥]Rel 𝐹[𝐴 / 𝑥]𝑤𝑦𝑧((𝑤𝐹𝑦𝑤𝐹𝑧) → 𝑦 = 𝑧)))
2 sbcrel 4454 . . . 4 (𝐴𝑉 → ([𝐴 / 𝑥]Rel 𝐹 ↔ Rel 𝐴 / 𝑥𝐹))
3 sbcal 2837 . . . . 5 ([𝐴 / 𝑥]𝑤𝑦𝑧((𝑤𝐹𝑦𝑤𝐹𝑧) → 𝑦 = 𝑧) ↔ ∀𝑤[𝐴 / 𝑥]𝑦𝑧((𝑤𝐹𝑦𝑤𝐹𝑧) → 𝑦 = 𝑧))
4 sbcal 2837 . . . . . . 7 ([𝐴 / 𝑥]𝑦𝑧((𝑤𝐹𝑦𝑤𝐹𝑧) → 𝑦 = 𝑧) ↔ ∀𝑦[𝐴 / 𝑥]𝑧((𝑤𝐹𝑦𝑤𝐹𝑧) → 𝑦 = 𝑧))
5 sbcal 2837 . . . . . . . . 9 ([𝐴 / 𝑥]𝑧((𝑤𝐹𝑦𝑤𝐹𝑧) → 𝑦 = 𝑧) ↔ ∀𝑧[𝐴 / 𝑥]((𝑤𝐹𝑦𝑤𝐹𝑧) → 𝑦 = 𝑧))
6 sbcimg 2827 . . . . . . . . . . 11 (𝐴𝑉 → ([𝐴 / 𝑥]((𝑤𝐹𝑦𝑤𝐹𝑧) → 𝑦 = 𝑧) ↔ ([𝐴 / 𝑥](𝑤𝐹𝑦𝑤𝐹𝑧) → [𝐴 / 𝑥]𝑦 = 𝑧)))
7 sbcan 2828 . . . . . . . . . . . . 13 ([𝐴 / 𝑥](𝑤𝐹𝑦𝑤𝐹𝑧) ↔ ([𝐴 / 𝑥]𝑤𝐹𝑦[𝐴 / 𝑥]𝑤𝐹𝑧))
8 sbcbrg 3841 . . . . . . . . . . . . . . 15 (𝐴𝑉 → ([𝐴 / 𝑥]𝑤𝐹𝑦𝐴 / 𝑥𝑤𝐴 / 𝑥𝐹𝐴 / 𝑥𝑦))
9 csbconstg 2892 . . . . . . . . . . . . . . . 16 (𝐴𝑉𝐴 / 𝑥𝑤 = 𝑤)
10 csbconstg 2892 . . . . . . . . . . . . . . . 16 (𝐴𝑉𝐴 / 𝑥𝑦 = 𝑦)
119, 10breq12d 3805 . . . . . . . . . . . . . . 15 (𝐴𝑉 → (𝐴 / 𝑥𝑤𝐴 / 𝑥𝐹𝐴 / 𝑥𝑦𝑤𝐴 / 𝑥𝐹𝑦))
128, 11bitrd 181 . . . . . . . . . . . . . 14 (𝐴𝑉 → ([𝐴 / 𝑥]𝑤𝐹𝑦𝑤𝐴 / 𝑥𝐹𝑦))
13 sbcbrg 3841 . . . . . . . . . . . . . . 15 (𝐴𝑉 → ([𝐴 / 𝑥]𝑤𝐹𝑧𝐴 / 𝑥𝑤𝐴 / 𝑥𝐹𝐴 / 𝑥𝑧))
14 csbconstg 2892 . . . . . . . . . . . . . . . 16 (𝐴𝑉𝐴 / 𝑥𝑧 = 𝑧)
159, 14breq12d 3805 . . . . . . . . . . . . . . 15 (𝐴𝑉 → (𝐴 / 𝑥𝑤𝐴 / 𝑥𝐹𝐴 / 𝑥𝑧𝑤𝐴 / 𝑥𝐹𝑧))
1613, 15bitrd 181 . . . . . . . . . . . . . 14 (𝐴𝑉 → ([𝐴 / 𝑥]𝑤𝐹𝑧𝑤𝐴 / 𝑥𝐹𝑧))
1712, 16anbi12d 450 . . . . . . . . . . . . 13 (𝐴𝑉 → (([𝐴 / 𝑥]𝑤𝐹𝑦[𝐴 / 𝑥]𝑤𝐹𝑧) ↔ (𝑤𝐴 / 𝑥𝐹𝑦𝑤𝐴 / 𝑥𝐹𝑧)))
187, 17syl5bb 185 . . . . . . . . . . . 12 (𝐴𝑉 → ([𝐴 / 𝑥](𝑤𝐹𝑦𝑤𝐹𝑧) ↔ (𝑤𝐴 / 𝑥𝐹𝑦𝑤𝐴 / 𝑥𝐹𝑧)))
19 sbcg 2855 . . . . . . . . . . . 12 (𝐴𝑉 → ([𝐴 / 𝑥]𝑦 = 𝑧𝑦 = 𝑧))
2018, 19imbi12d 227 . . . . . . . . . . 11 (𝐴𝑉 → (([𝐴 / 𝑥](𝑤𝐹𝑦𝑤𝐹𝑧) → [𝐴 / 𝑥]𝑦 = 𝑧) ↔ ((𝑤𝐴 / 𝑥𝐹𝑦𝑤𝐴 / 𝑥𝐹𝑧) → 𝑦 = 𝑧)))
216, 20bitrd 181 . . . . . . . . . 10 (𝐴𝑉 → ([𝐴 / 𝑥]((𝑤𝐹𝑦𝑤𝐹𝑧) → 𝑦 = 𝑧) ↔ ((𝑤𝐴 / 𝑥𝐹𝑦𝑤𝐴 / 𝑥𝐹𝑧) → 𝑦 = 𝑧)))
2221albidv 1721 . . . . . . . . 9 (𝐴𝑉 → (∀𝑧[𝐴 / 𝑥]((𝑤𝐹𝑦𝑤𝐹𝑧) → 𝑦 = 𝑧) ↔ ∀𝑧((𝑤𝐴 / 𝑥𝐹𝑦𝑤𝐴 / 𝑥𝐹𝑧) → 𝑦 = 𝑧)))
235, 22syl5bb 185 . . . . . . . 8 (𝐴𝑉 → ([𝐴 / 𝑥]𝑧((𝑤𝐹𝑦𝑤𝐹𝑧) → 𝑦 = 𝑧) ↔ ∀𝑧((𝑤𝐴 / 𝑥𝐹𝑦𝑤𝐴 / 𝑥𝐹𝑧) → 𝑦 = 𝑧)))
2423albidv 1721 . . . . . . 7 (𝐴𝑉 → (∀𝑦[𝐴 / 𝑥]𝑧((𝑤𝐹𝑦𝑤𝐹𝑧) → 𝑦 = 𝑧) ↔ ∀𝑦𝑧((𝑤𝐴 / 𝑥𝐹𝑦𝑤𝐴 / 𝑥𝐹𝑧) → 𝑦 = 𝑧)))
254, 24syl5bb 185 . . . . . 6 (𝐴𝑉 → ([𝐴 / 𝑥]𝑦𝑧((𝑤𝐹𝑦𝑤𝐹𝑧) → 𝑦 = 𝑧) ↔ ∀𝑦𝑧((𝑤𝐴 / 𝑥𝐹𝑦𝑤𝐴 / 𝑥𝐹𝑧) → 𝑦 = 𝑧)))
2625albidv 1721 . . . . 5 (𝐴𝑉 → (∀𝑤[𝐴 / 𝑥]𝑦𝑧((𝑤𝐹𝑦𝑤𝐹𝑧) → 𝑦 = 𝑧) ↔ ∀𝑤𝑦𝑧((𝑤𝐴 / 𝑥𝐹𝑦𝑤𝐴 / 𝑥𝐹𝑧) → 𝑦 = 𝑧)))
273, 26syl5bb 185 . . . 4 (𝐴𝑉 → ([𝐴 / 𝑥]𝑤𝑦𝑧((𝑤𝐹𝑦𝑤𝐹𝑧) → 𝑦 = 𝑧) ↔ ∀𝑤𝑦𝑧((𝑤𝐴 / 𝑥𝐹𝑦𝑤𝐴 / 𝑥𝐹𝑧) → 𝑦 = 𝑧)))
282, 27anbi12d 450 . . 3 (𝐴𝑉 → (([𝐴 / 𝑥]Rel 𝐹[𝐴 / 𝑥]𝑤𝑦𝑧((𝑤𝐹𝑦𝑤𝐹𝑧) → 𝑦 = 𝑧)) ↔ (Rel 𝐴 / 𝑥𝐹 ∧ ∀𝑤𝑦𝑧((𝑤𝐴 / 𝑥𝐹𝑦𝑤𝐴 / 𝑥𝐹𝑧) → 𝑦 = 𝑧))))
291, 28syl5bb 185 . 2 (𝐴𝑉 → ([𝐴 / 𝑥](Rel 𝐹 ∧ ∀𝑤𝑦𝑧((𝑤𝐹𝑦𝑤𝐹𝑧) → 𝑦 = 𝑧)) ↔ (Rel 𝐴 / 𝑥𝐹 ∧ ∀𝑤𝑦𝑧((𝑤𝐴 / 𝑥𝐹𝑦𝑤𝐴 / 𝑥𝐹𝑧) → 𝑦 = 𝑧))))
30 dffun2 4940 . . 3 (Fun 𝐹 ↔ (Rel 𝐹 ∧ ∀𝑤𝑦𝑧((𝑤𝐹𝑦𝑤𝐹𝑧) → 𝑦 = 𝑧)))
3130sbcbii 2845 . 2 ([𝐴 / 𝑥]Fun 𝐹[𝐴 / 𝑥](Rel 𝐹 ∧ ∀𝑤𝑦𝑧((𝑤𝐹𝑦𝑤𝐹𝑧) → 𝑦 = 𝑧)))
32 dffun2 4940 . 2 (Fun 𝐴 / 𝑥𝐹 ↔ (Rel 𝐴 / 𝑥𝐹 ∧ ∀𝑤𝑦𝑧((𝑤𝐴 / 𝑥𝐹𝑦𝑤𝐴 / 𝑥𝐹𝑧) → 𝑦 = 𝑧)))
3329, 31, 323bitr4g 216 1 (𝐴𝑉 → ([𝐴 / 𝑥]Fun 𝐹 ↔ Fun 𝐴 / 𝑥𝐹))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 101  wb 102  wal 1257  wcel 1409  [wsbc 2787  csb 2880   class class class wbr 3792  Rel wrel 4378  Fun wfun 4924
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3903  ax-pow 3955  ax-pr 3972
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-v 2576  df-sbc 2788  df-csb 2881  df-un 2950  df-in 2952  df-ss 2959  df-pw 3389  df-sn 3409  df-pr 3410  df-op 3412  df-br 3793  df-opab 3847  df-id 4058  df-rel 4380  df-cnv 4381  df-co 4382  df-fun 4932
This theorem is referenced by:  sbcfng  5072
  Copyright terms: Public domain W3C validator