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Theorem sbcfung 6375
 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 3824 . . 3 ([𝐴 / 𝑥](Rel 𝐹 ∧ ∀𝑤𝑦𝑧(𝑤𝐹𝑧𝑧 = 𝑦)) ↔ ([𝐴 / 𝑥]Rel 𝐹[𝐴 / 𝑥]𝑤𝑦𝑧(𝑤𝐹𝑧𝑧 = 𝑦)))
2 sbcrel 5653 . . . 4 (𝐴𝑉 → ([𝐴 / 𝑥]Rel 𝐹 ↔ Rel 𝐴 / 𝑥𝐹))
3 sbcal 3836 . . . . 5 ([𝐴 / 𝑥]𝑤𝑦𝑧(𝑤𝐹𝑧𝑧 = 𝑦) ↔ ∀𝑤[𝐴 / 𝑥]𝑦𝑧(𝑤𝐹𝑧𝑧 = 𝑦))
4 sbcex2 3837 . . . . . . 7 ([𝐴 / 𝑥]𝑦𝑧(𝑤𝐹𝑧𝑧 = 𝑦) ↔ ∃𝑦[𝐴 / 𝑥]𝑧(𝑤𝐹𝑧𝑧 = 𝑦))
5 sbcal 3836 . . . . . . . . 9 ([𝐴 / 𝑥]𝑧(𝑤𝐹𝑧𝑧 = 𝑦) ↔ ∀𝑧[𝐴 / 𝑥](𝑤𝐹𝑧𝑧 = 𝑦))
6 sbcimg 3823 . . . . . . . . . . 11 (𝐴𝑉 → ([𝐴 / 𝑥](𝑤𝐹𝑧𝑧 = 𝑦) ↔ ([𝐴 / 𝑥]𝑤𝐹𝑧[𝐴 / 𝑥]𝑧 = 𝑦)))
7 sbcbr123 5116 . . . . . . . . . . . . 13 ([𝐴 / 𝑥]𝑤𝐹𝑧𝐴 / 𝑥𝑤𝐴 / 𝑥𝐹𝐴 / 𝑥𝑧)
8 csbconstg 3905 . . . . . . . . . . . . . 14 (𝐴𝑉𝐴 / 𝑥𝑤 = 𝑤)
9 csbconstg 3905 . . . . . . . . . . . . . 14 (𝐴𝑉𝐴 / 𝑥𝑧 = 𝑧)
108, 9breq12d 5075 . . . . . . . . . . . . 13 (𝐴𝑉 → (𝐴 / 𝑥𝑤𝐴 / 𝑥𝐹𝐴 / 𝑥𝑧𝑤𝐴 / 𝑥𝐹𝑧))
117, 10syl5bb 284 . . . . . . . . . . . 12 (𝐴𝑉 → ([𝐴 / 𝑥]𝑤𝐹𝑧𝑤𝐴 / 𝑥𝐹𝑧))
12 sbcg 3850 . . . . . . . . . . . 12 (𝐴𝑉 → ([𝐴 / 𝑥]𝑧 = 𝑦𝑧 = 𝑦))
1311, 12imbi12d 346 . . . . . . . . . . 11 (𝐴𝑉 → (([𝐴 / 𝑥]𝑤𝐹𝑧[𝐴 / 𝑥]𝑧 = 𝑦) ↔ (𝑤𝐴 / 𝑥𝐹𝑧𝑧 = 𝑦)))
146, 13bitrd 280 . . . . . . . . . 10 (𝐴𝑉 → ([𝐴 / 𝑥](𝑤𝐹𝑧𝑧 = 𝑦) ↔ (𝑤𝐴 / 𝑥𝐹𝑧𝑧 = 𝑦)))
1514albidv 1914 . . . . . . . . 9 (𝐴𝑉 → (∀𝑧[𝐴 / 𝑥](𝑤𝐹𝑧𝑧 = 𝑦) ↔ ∀𝑧(𝑤𝐴 / 𝑥𝐹𝑧𝑧 = 𝑦)))
165, 15syl5bb 284 . . . . . . . 8 (𝐴𝑉 → ([𝐴 / 𝑥]𝑧(𝑤𝐹𝑧𝑧 = 𝑦) ↔ ∀𝑧(𝑤𝐴 / 𝑥𝐹𝑧𝑧 = 𝑦)))
1716exbidv 1915 . . . . . . 7 (𝐴𝑉 → (∃𝑦[𝐴 / 𝑥]𝑧(𝑤𝐹𝑧𝑧 = 𝑦) ↔ ∃𝑦𝑧(𝑤𝐴 / 𝑥𝐹𝑧𝑧 = 𝑦)))
184, 17syl5bb 284 . . . . . 6 (𝐴𝑉 → ([𝐴 / 𝑥]𝑦𝑧(𝑤𝐹𝑧𝑧 = 𝑦) ↔ ∃𝑦𝑧(𝑤𝐴 / 𝑥𝐹𝑧𝑧 = 𝑦)))
1918albidv 1914 . . . . 5 (𝐴𝑉 → (∀𝑤[𝐴 / 𝑥]𝑦𝑧(𝑤𝐹𝑧𝑧 = 𝑦) ↔ ∀𝑤𝑦𝑧(𝑤𝐴 / 𝑥𝐹𝑧𝑧 = 𝑦)))
203, 19syl5bb 284 . . . 4 (𝐴𝑉 → ([𝐴 / 𝑥]𝑤𝑦𝑧(𝑤𝐹𝑧𝑧 = 𝑦) ↔ ∀𝑤𝑦𝑧(𝑤𝐴 / 𝑥𝐹𝑧𝑧 = 𝑦)))
212, 20anbi12d 630 . . 3 (𝐴𝑉 → (([𝐴 / 𝑥]Rel 𝐹[𝐴 / 𝑥]𝑤𝑦𝑧(𝑤𝐹𝑧𝑧 = 𝑦)) ↔ (Rel 𝐴 / 𝑥𝐹 ∧ ∀𝑤𝑦𝑧(𝑤𝐴 / 𝑥𝐹𝑧𝑧 = 𝑦))))
221, 21syl5bb 284 . 2 (𝐴𝑉 → ([𝐴 / 𝑥](Rel 𝐹 ∧ ∀𝑤𝑦𝑧(𝑤𝐹𝑧𝑧 = 𝑦)) ↔ (Rel 𝐴 / 𝑥𝐹 ∧ ∀𝑤𝑦𝑧(𝑤𝐴 / 𝑥𝐹𝑧𝑧 = 𝑦))))
23 dffun3 6362 . . 3 (Fun 𝐹 ↔ (Rel 𝐹 ∧ ∀𝑤𝑦𝑧(𝑤𝐹𝑧𝑧 = 𝑦)))
2423sbcbii 3832 . 2 ([𝐴 / 𝑥]Fun 𝐹[𝐴 / 𝑥](Rel 𝐹 ∧ ∀𝑤𝑦𝑧(𝑤𝐹𝑧𝑧 = 𝑦)))
25 dffun3 6362 . 2 (Fun 𝐴 / 𝑥𝐹 ↔ (Rel 𝐴 / 𝑥𝐹 ∧ ∀𝑤𝑦𝑧(𝑤𝐴 / 𝑥𝐹𝑧𝑧 = 𝑦)))
2622, 24, 253bitr4g 315 1 (𝐴𝑉 → ([𝐴 / 𝑥]Fun 𝐹 ↔ Fun 𝐴 / 𝑥𝐹))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 207   ∧ wa 396  ∀wal 1528  ∃wex 1773   ∈ wcel 2106  [wsbc 3775  ⦋csb 3886   class class class wbr 5062  Rel wrel 5558  Fun wfun 6345 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2152  ax-12 2167  ax-ext 2796  ax-sep 5199  ax-nul 5206  ax-pr 5325 This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-fal 1543  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2615  df-eu 2649  df-clab 2803  df-cleq 2817  df-clel 2897  df-nfc 2967  df-ral 3147  df-rab 3151  df-v 3501  df-sbc 3776  df-csb 3887  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-nul 4295  df-if 4470  df-sn 4564  df-pr 4566  df-op 4570  df-br 5063  df-opab 5125  df-id 5458  df-rel 5560  df-cnv 5561  df-co 5562  df-fun 6353 This theorem is referenced by:  sbcfng  6507  esum2dlem  31238
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