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Theorem sbcfng 6343
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 6193 . . . 4 (𝐹 Fn 𝐴 ↔ (Fun 𝐹 ∧ dom 𝐹 = 𝐴))
21a1i 11 . . 3 (𝑋𝑉 → (𝐹 Fn 𝐴 ↔ (Fun 𝐹 ∧ dom 𝐹 = 𝐴)))
32sbcbidv 3733 . 2 (𝑋𝑉 → ([𝑋 / 𝑥]𝐹 Fn 𝐴[𝑋 / 𝑥](Fun 𝐹 ∧ dom 𝐹 = 𝐴)))
4 sbcfung 6214 . . . 4 (𝑋𝑉 → ([𝑋 / 𝑥]Fun 𝐹 ↔ Fun 𝑋 / 𝑥𝐹))
5 sbceqg 4248 . . . . 5 (𝑋𝑉 → ([𝑋 / 𝑥]dom 𝐹 = 𝐴𝑋 / 𝑥dom 𝐹 = 𝑋 / 𝑥𝐴))
6 csbdm 5617 . . . . . 6 𝑋 / 𝑥dom 𝐹 = dom 𝑋 / 𝑥𝐹
76eqeq1i 2783 . . . . 5 (𝑋 / 𝑥dom 𝐹 = 𝑋 / 𝑥𝐴 ↔ dom 𝑋 / 𝑥𝐹 = 𝑋 / 𝑥𝐴)
85, 7syl6bb 279 . . . 4 (𝑋𝑉 → ([𝑋 / 𝑥]dom 𝐹 = 𝐴 ↔ dom 𝑋 / 𝑥𝐹 = 𝑋 / 𝑥𝐴))
94, 8anbi12d 621 . . 3 (𝑋𝑉 → (([𝑋 / 𝑥]Fun 𝐹[𝑋 / 𝑥]dom 𝐹 = 𝐴) ↔ (Fun 𝑋 / 𝑥𝐹 ∧ dom 𝑋 / 𝑥𝐹 = 𝑋 / 𝑥𝐴)))
10 sbcan 3727 . . 3 ([𝑋 / 𝑥](Fun 𝐹 ∧ dom 𝐹 = 𝐴) ↔ ([𝑋 / 𝑥]Fun 𝐹[𝑋 / 𝑥]dom 𝐹 = 𝐴))
11 df-fn 6193 . . 3 (𝑋 / 𝑥𝐹 Fn 𝑋 / 𝑥𝐴 ↔ (Fun 𝑋 / 𝑥𝐹 ∧ dom 𝑋 / 𝑥𝐹 = 𝑋 / 𝑥𝐴))
129, 10, 113bitr4g 306 . 2 (𝑋𝑉 → ([𝑋 / 𝑥](Fun 𝐹 ∧ dom 𝐹 = 𝐴) ↔ 𝑋 / 𝑥𝐹 Fn 𝑋 / 𝑥𝐴))
133, 12bitrd 271 1 (𝑋𝑉 → ([𝑋 / 𝑥]𝐹 Fn 𝐴𝑋 / 𝑥𝐹 Fn 𝑋 / 𝑥𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 387   = wceq 1507  wcel 2050  [wsbc 3683  csb 3788  dom cdm 5408  Fun wfun 6184   Fn wfn 6185
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-13 2301  ax-ext 2750  ax-sep 5061  ax-nul 5068  ax-pr 5187
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3an 1070  df-tru 1510  df-fal 1520  df-ex 1743  df-nf 1747  df-sb 2016  df-mo 2547  df-eu 2583  df-clab 2759  df-cleq 2771  df-clel 2846  df-nfc 2918  df-ral 3093  df-rab 3097  df-v 3417  df-sbc 3684  df-csb 3789  df-dif 3834  df-un 3836  df-in 3838  df-ss 3845  df-nul 4181  df-if 4352  df-sn 4443  df-pr 4445  df-op 4449  df-br 4931  df-opab 4993  df-id 5313  df-rel 5415  df-cnv 5416  df-co 5417  df-dm 5418  df-fun 6192  df-fn 6193
This theorem is referenced by:  sbcfg  6344
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