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Theorem sbcfng 6488
Description: Distribute proper substitution through the function predicate with a domain. (Contributed by Alexander van der Vekens, 15-Jul-2018.)
Assertion
Ref Expression
sbcfng (𝑋𝑉 → ([𝑋 / 𝑥]𝐹 Fn 𝐴𝑋 / 𝑥𝐹 Fn 𝑋 / 𝑥𝐴))
Distinct variable groups:   𝑥,𝑉   𝑥,𝑋
Allowed substitution hints:   𝐴(𝑥)   𝐹(𝑥)

Proof of Theorem sbcfng
StepHypRef Expression
1 df-fn 6331 . . . 4 (𝐹 Fn 𝐴 ↔ (Fun 𝐹 ∧ dom 𝐹 = 𝐴))
21a1i 11 . . 3 (𝑋𝑉 → (𝐹 Fn 𝐴 ↔ (Fun 𝐹 ∧ dom 𝐹 = 𝐴)))
32sbcbidv 3777 . 2 (𝑋𝑉 → ([𝑋 / 𝑥]𝐹 Fn 𝐴[𝑋 / 𝑥](Fun 𝐹 ∧ dom 𝐹 = 𝐴)))
4 sbcfung 6352 . . . 4 (𝑋𝑉 → ([𝑋 / 𝑥]Fun 𝐹 ↔ Fun 𝑋 / 𝑥𝐹))
5 sbceqg 4320 . . . . 5 (𝑋𝑉 → ([𝑋 / 𝑥]dom 𝐹 = 𝐴𝑋 / 𝑥dom 𝐹 = 𝑋 / 𝑥𝐴))
6 csbdm 5734 . . . . . 6 𝑋 / 𝑥dom 𝐹 = dom 𝑋 / 𝑥𝐹
76eqeq1i 2806 . . . . 5 (𝑋 / 𝑥dom 𝐹 = 𝑋 / 𝑥𝐴 ↔ dom 𝑋 / 𝑥𝐹 = 𝑋 / 𝑥𝐴)
85, 7syl6bb 290 . . . 4 (𝑋𝑉 → ([𝑋 / 𝑥]dom 𝐹 = 𝐴 ↔ dom 𝑋 / 𝑥𝐹 = 𝑋 / 𝑥𝐴))
94, 8anbi12d 633 . . 3 (𝑋𝑉 → (([𝑋 / 𝑥]Fun 𝐹[𝑋 / 𝑥]dom 𝐹 = 𝐴) ↔ (Fun 𝑋 / 𝑥𝐹 ∧ dom 𝑋 / 𝑥𝐹 = 𝑋 / 𝑥𝐴)))
10 sbcan 3771 . . 3 ([𝑋 / 𝑥](Fun 𝐹 ∧ dom 𝐹 = 𝐴) ↔ ([𝑋 / 𝑥]Fun 𝐹[𝑋 / 𝑥]dom 𝐹 = 𝐴))
11 df-fn 6331 . . 3 (𝑋 / 𝑥𝐹 Fn 𝑋 / 𝑥𝐴 ↔ (Fun 𝑋 / 𝑥𝐹 ∧ dom 𝑋 / 𝑥𝐹 = 𝑋 / 𝑥𝐴))
129, 10, 113bitr4g 317 . 2 (𝑋𝑉 → ([𝑋 / 𝑥](Fun 𝐹 ∧ dom 𝐹 = 𝐴) ↔ 𝑋 / 𝑥𝐹 Fn 𝑋 / 𝑥𝐴))
133, 12bitrd 282 1 (𝑋𝑉 → ([𝑋 / 𝑥]𝐹 Fn 𝐴𝑋 / 𝑥𝐹 Fn 𝑋 / 𝑥𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399   = wceq 1538  wcel 2112  [wsbc 3723  csb 3831  dom cdm 5523  Fun wfun 6322   Fn wfn 6323
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-sep 5170  ax-nul 5177  ax-pr 5298
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ral 3114  df-v 3446  df-sbc 3724  df-csb 3832  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-sn 4529  df-pr 4531  df-op 4535  df-br 5034  df-opab 5096  df-id 5428  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-fun 6330  df-fn 6331
This theorem is referenced by:  sbcfg  6489
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