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Theorem sbcfung 5910
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 3476 . . 3 ([𝐴 / 𝑥](Rel 𝐹 ∧ ∀𝑤𝑦𝑧(𝑤𝐹𝑧𝑧 = 𝑦)) ↔ ([𝐴 / 𝑥]Rel 𝐹[𝐴 / 𝑥]𝑤𝑦𝑧(𝑤𝐹𝑧𝑧 = 𝑦)))
2 sbcrel 5203 . . . 4 (𝐴𝑉 → ([𝐴 / 𝑥]Rel 𝐹 ↔ Rel 𝐴 / 𝑥𝐹))
3 sbcal 3483 . . . . 5 ([𝐴 / 𝑥]𝑤𝑦𝑧(𝑤𝐹𝑧𝑧 = 𝑦) ↔ ∀𝑤[𝐴 / 𝑥]𝑦𝑧(𝑤𝐹𝑧𝑧 = 𝑦))
4 sbcex2 3484 . . . . . . 7 ([𝐴 / 𝑥]𝑦𝑧(𝑤𝐹𝑧𝑧 = 𝑦) ↔ ∃𝑦[𝐴 / 𝑥]𝑧(𝑤𝐹𝑧𝑧 = 𝑦))
5 sbcal 3483 . . . . . . . . 9 ([𝐴 / 𝑥]𝑧(𝑤𝐹𝑧𝑧 = 𝑦) ↔ ∀𝑧[𝐴 / 𝑥](𝑤𝐹𝑧𝑧 = 𝑦))
6 sbcimg 3475 . . . . . . . . . . 11 (𝐴𝑉 → ([𝐴 / 𝑥](𝑤𝐹𝑧𝑧 = 𝑦) ↔ ([𝐴 / 𝑥]𝑤𝐹𝑧[𝐴 / 𝑥]𝑧 = 𝑦)))
7 sbcbr123 4704 . . . . . . . . . . . . 13 ([𝐴 / 𝑥]𝑤𝐹𝑧𝐴 / 𝑥𝑤𝐴 / 𝑥𝐹𝐴 / 𝑥𝑧)
8 csbconstg 3544 . . . . . . . . . . . . . 14 (𝐴𝑉𝐴 / 𝑥𝑤 = 𝑤)
9 csbconstg 3544 . . . . . . . . . . . . . 14 (𝐴𝑉𝐴 / 𝑥𝑧 = 𝑧)
108, 9breq12d 4664 . . . . . . . . . . . . 13 (𝐴𝑉 → (𝐴 / 𝑥𝑤𝐴 / 𝑥𝐹𝐴 / 𝑥𝑧𝑤𝐴 / 𝑥𝐹𝑧))
117, 10syl5bb 272 . . . . . . . . . . . 12 (𝐴𝑉 → ([𝐴 / 𝑥]𝑤𝐹𝑧𝑤𝐴 / 𝑥𝐹𝑧))
12 sbcg 3501 . . . . . . . . . . . 12 (𝐴𝑉 → ([𝐴 / 𝑥]𝑧 = 𝑦𝑧 = 𝑦))
1311, 12imbi12d 334 . . . . . . . . . . 11 (𝐴𝑉 → (([𝐴 / 𝑥]𝑤𝐹𝑧[𝐴 / 𝑥]𝑧 = 𝑦) ↔ (𝑤𝐴 / 𝑥𝐹𝑧𝑧 = 𝑦)))
146, 13bitrd 268 . . . . . . . . . 10 (𝐴𝑉 → ([𝐴 / 𝑥](𝑤𝐹𝑧𝑧 = 𝑦) ↔ (𝑤𝐴 / 𝑥𝐹𝑧𝑧 = 𝑦)))
1514albidv 1848 . . . . . . . . 9 (𝐴𝑉 → (∀𝑧[𝐴 / 𝑥](𝑤𝐹𝑧𝑧 = 𝑦) ↔ ∀𝑧(𝑤𝐴 / 𝑥𝐹𝑧𝑧 = 𝑦)))
165, 15syl5bb 272 . . . . . . . 8 (𝐴𝑉 → ([𝐴 / 𝑥]𝑧(𝑤𝐹𝑧𝑧 = 𝑦) ↔ ∀𝑧(𝑤𝐴 / 𝑥𝐹𝑧𝑧 = 𝑦)))
1716exbidv 1849 . . . . . . 7 (𝐴𝑉 → (∃𝑦[𝐴 / 𝑥]𝑧(𝑤𝐹𝑧𝑧 = 𝑦) ↔ ∃𝑦𝑧(𝑤𝐴 / 𝑥𝐹𝑧𝑧 = 𝑦)))
184, 17syl5bb 272 . . . . . 6 (𝐴𝑉 → ([𝐴 / 𝑥]𝑦𝑧(𝑤𝐹𝑧𝑧 = 𝑦) ↔ ∃𝑦𝑧(𝑤𝐴 / 𝑥𝐹𝑧𝑧 = 𝑦)))
1918albidv 1848 . . . . 5 (𝐴𝑉 → (∀𝑤[𝐴 / 𝑥]𝑦𝑧(𝑤𝐹𝑧𝑧 = 𝑦) ↔ ∀𝑤𝑦𝑧(𝑤𝐴 / 𝑥𝐹𝑧𝑧 = 𝑦)))
203, 19syl5bb 272 . . . 4 (𝐴𝑉 → ([𝐴 / 𝑥]𝑤𝑦𝑧(𝑤𝐹𝑧𝑧 = 𝑦) ↔ ∀𝑤𝑦𝑧(𝑤𝐴 / 𝑥𝐹𝑧𝑧 = 𝑦)))
212, 20anbi12d 747 . . 3 (𝐴𝑉 → (([𝐴 / 𝑥]Rel 𝐹[𝐴 / 𝑥]𝑤𝑦𝑧(𝑤𝐹𝑧𝑧 = 𝑦)) ↔ (Rel 𝐴 / 𝑥𝐹 ∧ ∀𝑤𝑦𝑧(𝑤𝐴 / 𝑥𝐹𝑧𝑧 = 𝑦))))
221, 21syl5bb 272 . 2 (𝐴𝑉 → ([𝐴 / 𝑥](Rel 𝐹 ∧ ∀𝑤𝑦𝑧(𝑤𝐹𝑧𝑧 = 𝑦)) ↔ (Rel 𝐴 / 𝑥𝐹 ∧ ∀𝑤𝑦𝑧(𝑤𝐴 / 𝑥𝐹𝑧𝑧 = 𝑦))))
23 dffun3 5897 . . 3 (Fun 𝐹 ↔ (Rel 𝐹 ∧ ∀𝑤𝑦𝑧(𝑤𝐹𝑧𝑧 = 𝑦)))
2423sbcbii 3489 . 2 ([𝐴 / 𝑥]Fun 𝐹[𝐴 / 𝑥](Rel 𝐹 ∧ ∀𝑤𝑦𝑧(𝑤𝐹𝑧𝑧 = 𝑦)))
25 dffun3 5897 . 2 (Fun 𝐴 / 𝑥𝐹 ↔ (Rel 𝐴 / 𝑥𝐹 ∧ ∀𝑤𝑦𝑧(𝑤𝐴 / 𝑥𝐹𝑧𝑧 = 𝑦)))
2622, 24, 253bitr4g 303 1 (𝐴𝑉 → ([𝐴 / 𝑥]Fun 𝐹 ↔ Fun 𝐴 / 𝑥𝐹))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384  wal 1480  wex 1703  wcel 1989  [wsbc 3433  csb 3531   class class class wbr 4651  Rel wrel 5117  Fun wfun 5880
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1721  ax-4 1736  ax-5 1838  ax-6 1887  ax-7 1934  ax-9 1998  ax-10 2018  ax-11 2033  ax-12 2046  ax-13 2245  ax-ext 2601  ax-sep 4779  ax-nul 4787  ax-pr 4904
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1485  df-fal 1488  df-ex 1704  df-nf 1709  df-sb 1880  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2752  df-ral 2916  df-rab 2920  df-v 3200  df-sbc 3434  df-csb 3532  df-dif 3575  df-un 3577  df-in 3579  df-ss 3586  df-nul 3914  df-if 4085  df-sn 4176  df-pr 4178  df-op 4182  df-br 4652  df-opab 4711  df-id 5022  df-rel 5119  df-cnv 5120  df-co 5121  df-fun 5888
This theorem is referenced by:  sbcfng  6040  esum2dlem  30139
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