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Theorem sbcfung 6373
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 3737 . . 3 ([𝐴 / 𝑥](Rel 𝐹 ∧ ∀𝑤𝑦𝑧(𝑤𝐹𝑧𝑧 = 𝑦)) ↔ ([𝐴 / 𝑥]Rel 𝐹[𝐴 / 𝑥]𝑤𝑦𝑧(𝑤𝐹𝑧𝑧 = 𝑦)))
2 sbcrel 5636 . . . 4 (𝐴𝑉 → ([𝐴 / 𝑥]Rel 𝐹 ↔ Rel 𝐴 / 𝑥𝐹))
3 sbcal 3749 . . . . 5 ([𝐴 / 𝑥]𝑤𝑦𝑧(𝑤𝐹𝑧𝑧 = 𝑦) ↔ ∀𝑤[𝐴 / 𝑥]𝑦𝑧(𝑤𝐹𝑧𝑧 = 𝑦))
4 sbcex2 3750 . . . . . . 7 ([𝐴 / 𝑥]𝑦𝑧(𝑤𝐹𝑧𝑧 = 𝑦) ↔ ∃𝑦[𝐴 / 𝑥]𝑧(𝑤𝐹𝑧𝑧 = 𝑦))
5 sbcal 3749 . . . . . . . . 9 ([𝐴 / 𝑥]𝑧(𝑤𝐹𝑧𝑧 = 𝑦) ↔ ∀𝑧[𝐴 / 𝑥](𝑤𝐹𝑧𝑧 = 𝑦))
6 sbcimg 3736 . . . . . . . . . . 11 (𝐴𝑉 → ([𝐴 / 𝑥](𝑤𝐹𝑧𝑧 = 𝑦) ↔ ([𝐴 / 𝑥]𝑤𝐹𝑧[𝐴 / 𝑥]𝑧 = 𝑦)))
7 sbcbr123 5094 . . . . . . . . . . . . 13 ([𝐴 / 𝑥]𝑤𝐹𝑧𝐴 / 𝑥𝑤𝐴 / 𝑥𝐹𝐴 / 𝑥𝑧)
8 csbconstg 3819 . . . . . . . . . . . . . 14 (𝐴𝑉𝐴 / 𝑥𝑤 = 𝑤)
9 csbconstg 3819 . . . . . . . . . . . . . 14 (𝐴𝑉𝐴 / 𝑥𝑧 = 𝑧)
108, 9breq12d 5053 . . . . . . . . . . . . 13 (𝐴𝑉 → (𝐴 / 𝑥𝑤𝐴 / 𝑥𝐹𝐴 / 𝑥𝑧𝑤𝐴 / 𝑥𝐹𝑧))
117, 10syl5bb 286 . . . . . . . . . . . 12 (𝐴𝑉 → ([𝐴 / 𝑥]𝑤𝐹𝑧𝑤𝐴 / 𝑥𝐹𝑧))
12 sbcg 3763 . . . . . . . . . . . 12 (𝐴𝑉 → ([𝐴 / 𝑥]𝑧 = 𝑦𝑧 = 𝑦))
1311, 12imbi12d 348 . . . . . . . . . . 11 (𝐴𝑉 → (([𝐴 / 𝑥]𝑤𝐹𝑧[𝐴 / 𝑥]𝑧 = 𝑦) ↔ (𝑤𝐴 / 𝑥𝐹𝑧𝑧 = 𝑦)))
146, 13bitrd 282 . . . . . . . . . 10 (𝐴𝑉 → ([𝐴 / 𝑥](𝑤𝐹𝑧𝑧 = 𝑦) ↔ (𝑤𝐴 / 𝑥𝐹𝑧𝑧 = 𝑦)))
1514albidv 1927 . . . . . . . . 9 (𝐴𝑉 → (∀𝑧[𝐴 / 𝑥](𝑤𝐹𝑧𝑧 = 𝑦) ↔ ∀𝑧(𝑤𝐴 / 𝑥𝐹𝑧𝑧 = 𝑦)))
165, 15syl5bb 286 . . . . . . . 8 (𝐴𝑉 → ([𝐴 / 𝑥]𝑧(𝑤𝐹𝑧𝑧 = 𝑦) ↔ ∀𝑧(𝑤𝐴 / 𝑥𝐹𝑧𝑧 = 𝑦)))
1716exbidv 1928 . . . . . . 7 (𝐴𝑉 → (∃𝑦[𝐴 / 𝑥]𝑧(𝑤𝐹𝑧𝑧 = 𝑦) ↔ ∃𝑦𝑧(𝑤𝐴 / 𝑥𝐹𝑧𝑧 = 𝑦)))
184, 17syl5bb 286 . . . . . 6 (𝐴𝑉 → ([𝐴 / 𝑥]𝑦𝑧(𝑤𝐹𝑧𝑧 = 𝑦) ↔ ∃𝑦𝑧(𝑤𝐴 / 𝑥𝐹𝑧𝑧 = 𝑦)))
1918albidv 1927 . . . . 5 (𝐴𝑉 → (∀𝑤[𝐴 / 𝑥]𝑦𝑧(𝑤𝐹𝑧𝑧 = 𝑦) ↔ ∀𝑤𝑦𝑧(𝑤𝐴 / 𝑥𝐹𝑧𝑧 = 𝑦)))
203, 19syl5bb 286 . . . 4 (𝐴𝑉 → ([𝐴 / 𝑥]𝑤𝑦𝑧(𝑤𝐹𝑧𝑧 = 𝑦) ↔ ∀𝑤𝑦𝑧(𝑤𝐴 / 𝑥𝐹𝑧𝑧 = 𝑦)))
212, 20anbi12d 634 . . 3 (𝐴𝑉 → (([𝐴 / 𝑥]Rel 𝐹[𝐴 / 𝑥]𝑤𝑦𝑧(𝑤𝐹𝑧𝑧 = 𝑦)) ↔ (Rel 𝐴 / 𝑥𝐹 ∧ ∀𝑤𝑦𝑧(𝑤𝐴 / 𝑥𝐹𝑧𝑧 = 𝑦))))
221, 21syl5bb 286 . 2 (𝐴𝑉 → ([𝐴 / 𝑥](Rel 𝐹 ∧ ∀𝑤𝑦𝑧(𝑤𝐹𝑧𝑧 = 𝑦)) ↔ (Rel 𝐴 / 𝑥𝐹 ∧ ∀𝑤𝑦𝑧(𝑤𝐴 / 𝑥𝐹𝑧𝑧 = 𝑦))))
23 dffun3 6360 . . 3 (Fun 𝐹 ↔ (Rel 𝐹 ∧ ∀𝑤𝑦𝑧(𝑤𝐹𝑧𝑧 = 𝑦)))
2423sbcbii 3745 . 2 ([𝐴 / 𝑥]Fun 𝐹[𝐴 / 𝑥](Rel 𝐹 ∧ ∀𝑤𝑦𝑧(𝑤𝐹𝑧𝑧 = 𝑦)))
25 dffun3 6360 . 2 (Fun 𝐴 / 𝑥𝐹 ↔ (Rel 𝐴 / 𝑥𝐹 ∧ ∀𝑤𝑦𝑧(𝑤𝐴 / 𝑥𝐹𝑧𝑧 = 𝑦)))
2622, 24, 253bitr4g 317 1 (𝐴𝑉 → ([𝐴 / 𝑥]Fun 𝐹 ↔ Fun 𝐴 / 𝑥𝐹))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399  wal 1540  wex 1786  wcel 2114  [wsbc 3685  csb 3800   class class class wbr 5040  Rel wrel 5540  Fun wfun 6343
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2162  ax-12 2179  ax-ext 2711  ax-sep 5177  ax-nul 5184  ax-pr 5306
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2075  df-mo 2541  df-eu 2571  df-clab 2718  df-cleq 2731  df-clel 2812  df-nfc 2882  df-ral 3059  df-v 3402  df-sbc 3686  df-csb 3801  df-dif 3856  df-un 3858  df-in 3860  df-ss 3870  df-nul 4222  df-if 4425  df-sn 4527  df-pr 4529  df-op 4533  df-br 5041  df-opab 5103  df-id 5439  df-rel 5542  df-cnv 5543  df-co 5544  df-fun 6351
This theorem is referenced by:  sbcfng  6511  esum2dlem  31642
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