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Theorem sbcfng 6714
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 6546 . . . 4 (𝐹 Fn 𝐴 ↔ (Fun 𝐹 ∧ dom 𝐹 = 𝐴))
21a1i 11 . . 3 (𝑋𝑉 → (𝐹 Fn 𝐴 ↔ (Fun 𝐹 ∧ dom 𝐹 = 𝐴)))
32sbcbidv 3836 . 2 (𝑋𝑉 → ([𝑋 / 𝑥]𝐹 Fn 𝐴[𝑋 / 𝑥](Fun 𝐹 ∧ dom 𝐹 = 𝐴)))
4 sbcfung 6572 . . . 4 (𝑋𝑉 → ([𝑋 / 𝑥]Fun 𝐹 ↔ Fun 𝑋 / 𝑥𝐹))
5 sbceqg 4409 . . . . 5 (𝑋𝑉 → ([𝑋 / 𝑥]dom 𝐹 = 𝐴𝑋 / 𝑥dom 𝐹 = 𝑋 / 𝑥𝐴))
6 csbdm 5897 . . . . . 6 𝑋 / 𝑥dom 𝐹 = dom 𝑋 / 𝑥𝐹
76eqeq1i 2736 . . . . 5 (𝑋 / 𝑥dom 𝐹 = 𝑋 / 𝑥𝐴 ↔ dom 𝑋 / 𝑥𝐹 = 𝑋 / 𝑥𝐴)
85, 7bitrdi 287 . . . 4 (𝑋𝑉 → ([𝑋 / 𝑥]dom 𝐹 = 𝐴 ↔ dom 𝑋 / 𝑥𝐹 = 𝑋 / 𝑥𝐴))
94, 8anbi12d 630 . . 3 (𝑋𝑉 → (([𝑋 / 𝑥]Fun 𝐹[𝑋 / 𝑥]dom 𝐹 = 𝐴) ↔ (Fun 𝑋 / 𝑥𝐹 ∧ dom 𝑋 / 𝑥𝐹 = 𝑋 / 𝑥𝐴)))
10 sbcan 3829 . . 3 ([𝑋 / 𝑥](Fun 𝐹 ∧ dom 𝐹 = 𝐴) ↔ ([𝑋 / 𝑥]Fun 𝐹[𝑋 / 𝑥]dom 𝐹 = 𝐴))
11 df-fn 6546 . . 3 (𝑋 / 𝑥𝐹 Fn 𝑋 / 𝑥𝐴 ↔ (Fun 𝑋 / 𝑥𝐹 ∧ dom 𝑋 / 𝑥𝐹 = 𝑋 / 𝑥𝐴))
129, 10, 113bitr4g 314 . 2 (𝑋𝑉 → ([𝑋 / 𝑥](Fun 𝐹 ∧ dom 𝐹 = 𝐴) ↔ 𝑋 / 𝑥𝐹 Fn 𝑋 / 𝑥𝐴))
133, 12bitrd 279 1 (𝑋𝑉 → ([𝑋 / 𝑥]𝐹 Fn 𝐴𝑋 / 𝑥𝐹 Fn 𝑋 / 𝑥𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 395   = wceq 1540  wcel 2105  [wsbc 3777  csb 3893  dom cdm 5676  Fun wfun 6537   Fn wfn 6538
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-sep 5299  ax-nul 5306  ax-pr 5427
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ral 3061  df-rex 3070  df-rab 3432  df-v 3475  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-br 5149  df-opab 5211  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-fun 6545  df-fn 6546
This theorem is referenced by:  sbcfg  6715
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