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Theorem sbcfung 6526
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 3792 . . 3 ([𝐴 / 𝑥](Rel 𝐹 ∧ ∀𝑤𝑦𝑧(𝑤𝐹𝑧𝑧 = 𝑦)) ↔ ([𝐴 / 𝑥]Rel 𝐹[𝐴 / 𝑥]𝑤𝑦𝑧(𝑤𝐹𝑧𝑧 = 𝑦)))
2 sbcrel 5737 . . . 4 (𝐴𝑉 → ([𝐴 / 𝑥]Rel 𝐹 ↔ Rel 𝐴 / 𝑥𝐹))
3 sbcal 3804 . . . . 5 ([𝐴 / 𝑥]𝑤𝑦𝑧(𝑤𝐹𝑧𝑧 = 𝑦) ↔ ∀𝑤[𝐴 / 𝑥]𝑦𝑧(𝑤𝐹𝑧𝑧 = 𝑦))
4 sbcex2 3805 . . . . . . 7 ([𝐴 / 𝑥]𝑦𝑧(𝑤𝐹𝑧𝑧 = 𝑦) ↔ ∃𝑦[𝐴 / 𝑥]𝑧(𝑤𝐹𝑧𝑧 = 𝑦))
5 sbcal 3804 . . . . . . . . 9 ([𝐴 / 𝑥]𝑧(𝑤𝐹𝑧𝑧 = 𝑦) ↔ ∀𝑧[𝐴 / 𝑥](𝑤𝐹𝑧𝑧 = 𝑦))
6 sbcimg 3791 . . . . . . . . . . 11 (𝐴𝑉 → ([𝐴 / 𝑥](𝑤𝐹𝑧𝑧 = 𝑦) ↔ ([𝐴 / 𝑥]𝑤𝐹𝑧[𝐴 / 𝑥]𝑧 = 𝑦)))
7 sbcbr123 5160 . . . . . . . . . . . . 13 ([𝐴 / 𝑥]𝑤𝐹𝑧𝐴 / 𝑥𝑤𝐴 / 𝑥𝐹𝐴 / 𝑥𝑧)
8 csbconstg 3875 . . . . . . . . . . . . . 14 (𝐴𝑉𝐴 / 𝑥𝑤 = 𝑤)
9 csbconstg 3875 . . . . . . . . . . . . . 14 (𝐴𝑉𝐴 / 𝑥𝑧 = 𝑧)
108, 9breq12d 5119 . . . . . . . . . . . . 13 (𝐴𝑉 → (𝐴 / 𝑥𝑤𝐴 / 𝑥𝐹𝐴 / 𝑥𝑧𝑤𝐴 / 𝑥𝐹𝑧))
117, 10bitrid 283 . . . . . . . . . . . 12 (𝐴𝑉 → ([𝐴 / 𝑥]𝑤𝐹𝑧𝑤𝐴 / 𝑥𝐹𝑧))
12 sbcg 3819 . . . . . . . . . . . 12 (𝐴𝑉 → ([𝐴 / 𝑥]𝑧 = 𝑦𝑧 = 𝑦))
1311, 12imbi12d 345 . . . . . . . . . . 11 (𝐴𝑉 → (([𝐴 / 𝑥]𝑤𝐹𝑧[𝐴 / 𝑥]𝑧 = 𝑦) ↔ (𝑤𝐴 / 𝑥𝐹𝑧𝑧 = 𝑦)))
146, 13bitrd 279 . . . . . . . . . 10 (𝐴𝑉 → ([𝐴 / 𝑥](𝑤𝐹𝑧𝑧 = 𝑦) ↔ (𝑤𝐴 / 𝑥𝐹𝑧𝑧 = 𝑦)))
1514albidv 1924 . . . . . . . . 9 (𝐴𝑉 → (∀𝑧[𝐴 / 𝑥](𝑤𝐹𝑧𝑧 = 𝑦) ↔ ∀𝑧(𝑤𝐴 / 𝑥𝐹𝑧𝑧 = 𝑦)))
165, 15bitrid 283 . . . . . . . 8 (𝐴𝑉 → ([𝐴 / 𝑥]𝑧(𝑤𝐹𝑧𝑧 = 𝑦) ↔ ∀𝑧(𝑤𝐴 / 𝑥𝐹𝑧𝑧 = 𝑦)))
1716exbidv 1925 . . . . . . 7 (𝐴𝑉 → (∃𝑦[𝐴 / 𝑥]𝑧(𝑤𝐹𝑧𝑧 = 𝑦) ↔ ∃𝑦𝑧(𝑤𝐴 / 𝑥𝐹𝑧𝑧 = 𝑦)))
184, 17bitrid 283 . . . . . 6 (𝐴𝑉 → ([𝐴 / 𝑥]𝑦𝑧(𝑤𝐹𝑧𝑧 = 𝑦) ↔ ∃𝑦𝑧(𝑤𝐴 / 𝑥𝐹𝑧𝑧 = 𝑦)))
1918albidv 1924 . . . . 5 (𝐴𝑉 → (∀𝑤[𝐴 / 𝑥]𝑦𝑧(𝑤𝐹𝑧𝑧 = 𝑦) ↔ ∀𝑤𝑦𝑧(𝑤𝐴 / 𝑥𝐹𝑧𝑧 = 𝑦)))
203, 19bitrid 283 . . . 4 (𝐴𝑉 → ([𝐴 / 𝑥]𝑤𝑦𝑧(𝑤𝐹𝑧𝑧 = 𝑦) ↔ ∀𝑤𝑦𝑧(𝑤𝐴 / 𝑥𝐹𝑧𝑧 = 𝑦)))
212, 20anbi12d 632 . . 3 (𝐴𝑉 → (([𝐴 / 𝑥]Rel 𝐹[𝐴 / 𝑥]𝑤𝑦𝑧(𝑤𝐹𝑧𝑧 = 𝑦)) ↔ (Rel 𝐴 / 𝑥𝐹 ∧ ∀𝑤𝑦𝑧(𝑤𝐴 / 𝑥𝐹𝑧𝑧 = 𝑦))))
221, 21bitrid 283 . 2 (𝐴𝑉 → ([𝐴 / 𝑥](Rel 𝐹 ∧ ∀𝑤𝑦𝑧(𝑤𝐹𝑧𝑧 = 𝑦)) ↔ (Rel 𝐴 / 𝑥𝐹 ∧ ∀𝑤𝑦𝑧(𝑤𝐴 / 𝑥𝐹𝑧𝑧 = 𝑦))))
23 dffun3 6511 . . 3 (Fun 𝐹 ↔ (Rel 𝐹 ∧ ∀𝑤𝑦𝑧(𝑤𝐹𝑧𝑧 = 𝑦)))
2423sbcbii 3800 . 2 ([𝐴 / 𝑥]Fun 𝐹[𝐴 / 𝑥](Rel 𝐹 ∧ ∀𝑤𝑦𝑧(𝑤𝐹𝑧𝑧 = 𝑦)))
25 dffun3 6511 . 2 (Fun 𝐴 / 𝑥𝐹 ↔ (Rel 𝐴 / 𝑥𝐹 ∧ ∀𝑤𝑦𝑧(𝑤𝐴 / 𝑥𝐹𝑧𝑧 = 𝑦)))
2622, 24, 253bitr4g 314 1 (𝐴𝑉 → ([𝐴 / 𝑥]Fun 𝐹 ↔ Fun 𝐴 / 𝑥𝐹))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 397  wal 1540  wex 1782  wcel 2107  [wsbc 3740  csb 3856   class class class wbr 5106  Rel wrel 5639  Fun wfun 6491
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5257  ax-nul 5264  ax-pr 5385
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3446  df-sbc 3741  df-csb 3857  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-sn 4588  df-pr 4590  df-op 4594  df-br 5107  df-opab 5169  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-fun 6499
This theorem is referenced by:  sbcfng  6666  esum2dlem  32748
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