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Theorem sbcfung 5827
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 3445 . . 3 ([𝐴 / 𝑥](Rel 𝐹 ∧ ∀𝑤𝑦𝑧(𝑤𝐹𝑧𝑧 = 𝑦)) ↔ ([𝐴 / 𝑥]Rel 𝐹[𝐴 / 𝑥]𝑤𝑦𝑧(𝑤𝐹𝑧𝑧 = 𝑦)))
2 sbcrel 5128 . . . 4 (𝐴𝑉 → ([𝐴 / 𝑥]Rel 𝐹 ↔ Rel 𝐴 / 𝑥𝐹))
3 sbcal 3452 . . . . 5 ([𝐴 / 𝑥]𝑤𝑦𝑧(𝑤𝐹𝑧𝑧 = 𝑦) ↔ ∀𝑤[𝐴 / 𝑥]𝑦𝑧(𝑤𝐹𝑧𝑧 = 𝑦))
4 sbcex2 3453 . . . . . . 7 ([𝐴 / 𝑥]𝑦𝑧(𝑤𝐹𝑧𝑧 = 𝑦) ↔ ∃𝑦[𝐴 / 𝑥]𝑧(𝑤𝐹𝑧𝑧 = 𝑦))
5 sbcal 3452 . . . . . . . . 9 ([𝐴 / 𝑥]𝑧(𝑤𝐹𝑧𝑧 = 𝑦) ↔ ∀𝑧[𝐴 / 𝑥](𝑤𝐹𝑧𝑧 = 𝑦))
6 sbcimg 3444 . . . . . . . . . . 11 (𝐴𝑉 → ([𝐴 / 𝑥](𝑤𝐹𝑧𝑧 = 𝑦) ↔ ([𝐴 / 𝑥]𝑤𝐹𝑧[𝐴 / 𝑥]𝑧 = 𝑦)))
7 sbcbr123 4636 . . . . . . . . . . . . 13 ([𝐴 / 𝑥]𝑤𝐹𝑧𝐴 / 𝑥𝑤𝐴 / 𝑥𝐹𝐴 / 𝑥𝑧)
8 csbconstg 3512 . . . . . . . . . . . . . 14 (𝐴𝑉𝐴 / 𝑥𝑤 = 𝑤)
9 csbconstg 3512 . . . . . . . . . . . . . 14 (𝐴𝑉𝐴 / 𝑥𝑧 = 𝑧)
108, 9breq12d 4596 . . . . . . . . . . . . 13 (𝐴𝑉 → (𝐴 / 𝑥𝑤𝐴 / 𝑥𝐹𝐴 / 𝑥𝑧𝑤𝐴 / 𝑥𝐹𝑧))
117, 10syl5bb 271 . . . . . . . . . . . 12 (𝐴𝑉 → ([𝐴 / 𝑥]𝑤𝐹𝑧𝑤𝐴 / 𝑥𝐹𝑧))
12 sbcg 3470 . . . . . . . . . . . 12 (𝐴𝑉 → ([𝐴 / 𝑥]𝑧 = 𝑦𝑧 = 𝑦))
1311, 12imbi12d 333 . . . . . . . . . . 11 (𝐴𝑉 → (([𝐴 / 𝑥]𝑤𝐹𝑧[𝐴 / 𝑥]𝑧 = 𝑦) ↔ (𝑤𝐴 / 𝑥𝐹𝑧𝑧 = 𝑦)))
146, 13bitrd 267 . . . . . . . . . 10 (𝐴𝑉 → ([𝐴 / 𝑥](𝑤𝐹𝑧𝑧 = 𝑦) ↔ (𝑤𝐴 / 𝑥𝐹𝑧𝑧 = 𝑦)))
1514albidv 1836 . . . . . . . . 9 (𝐴𝑉 → (∀𝑧[𝐴 / 𝑥](𝑤𝐹𝑧𝑧 = 𝑦) ↔ ∀𝑧(𝑤𝐴 / 𝑥𝐹𝑧𝑧 = 𝑦)))
165, 15syl5bb 271 . . . . . . . 8 (𝐴𝑉 → ([𝐴 / 𝑥]𝑧(𝑤𝐹𝑧𝑧 = 𝑦) ↔ ∀𝑧(𝑤𝐴 / 𝑥𝐹𝑧𝑧 = 𝑦)))
1716exbidv 1837 . . . . . . 7 (𝐴𝑉 → (∃𝑦[𝐴 / 𝑥]𝑧(𝑤𝐹𝑧𝑧 = 𝑦) ↔ ∃𝑦𝑧(𝑤𝐴 / 𝑥𝐹𝑧𝑧 = 𝑦)))
184, 17syl5bb 271 . . . . . 6 (𝐴𝑉 → ([𝐴 / 𝑥]𝑦𝑧(𝑤𝐹𝑧𝑧 = 𝑦) ↔ ∃𝑦𝑧(𝑤𝐴 / 𝑥𝐹𝑧𝑧 = 𝑦)))
1918albidv 1836 . . . . 5 (𝐴𝑉 → (∀𝑤[𝐴 / 𝑥]𝑦𝑧(𝑤𝐹𝑧𝑧 = 𝑦) ↔ ∀𝑤𝑦𝑧(𝑤𝐴 / 𝑥𝐹𝑧𝑧 = 𝑦)))
203, 19syl5bb 271 . . . 4 (𝐴𝑉 → ([𝐴 / 𝑥]𝑤𝑦𝑧(𝑤𝐹𝑧𝑧 = 𝑦) ↔ ∀𝑤𝑦𝑧(𝑤𝐴 / 𝑥𝐹𝑧𝑧 = 𝑦)))
212, 20anbi12d 743 . . 3 (𝐴𝑉 → (([𝐴 / 𝑥]Rel 𝐹[𝐴 / 𝑥]𝑤𝑦𝑧(𝑤𝐹𝑧𝑧 = 𝑦)) ↔ (Rel 𝐴 / 𝑥𝐹 ∧ ∀𝑤𝑦𝑧(𝑤𝐴 / 𝑥𝐹𝑧𝑧 = 𝑦))))
221, 21syl5bb 271 . 2 (𝐴𝑉 → ([𝐴 / 𝑥](Rel 𝐹 ∧ ∀𝑤𝑦𝑧(𝑤𝐹𝑧𝑧 = 𝑦)) ↔ (Rel 𝐴 / 𝑥𝐹 ∧ ∀𝑤𝑦𝑧(𝑤𝐴 / 𝑥𝐹𝑧𝑧 = 𝑦))))
23 dffun3 5815 . . 3 (Fun 𝐹 ↔ (Rel 𝐹 ∧ ∀𝑤𝑦𝑧(𝑤𝐹𝑧𝑧 = 𝑦)))
2423sbcbii 3458 . 2 ([𝐴 / 𝑥]Fun 𝐹[𝐴 / 𝑥](Rel 𝐹 ∧ ∀𝑤𝑦𝑧(𝑤𝐹𝑧𝑧 = 𝑦)))
25 dffun3 5815 . 2 (Fun 𝐴 / 𝑥𝐹 ↔ (Rel 𝐴 / 𝑥𝐹 ∧ ∀𝑤𝑦𝑧(𝑤𝐴 / 𝑥𝐹𝑧𝑧 = 𝑦)))
2622, 24, 253bitr4g 302 1 (𝐴𝑉 → ([𝐴 / 𝑥]Fun 𝐹 ↔ Fun 𝐴 / 𝑥𝐹))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 195  wa 383  wal 1473  wex 1695  wcel 1977  [wsbc 3402  csb 3499   class class class wbr 4583  Rel wrel 5043  Fun wfun 5798
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-fal 1481  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-br 4584  df-opab 4644  df-id 4953  df-rel 5045  df-cnv 5046  df-co 5047  df-fun 5806
This theorem is referenced by:  sbcfng  5955  esum2dlem  29481
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