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Theorem sbcfng 4987
Description: Distribute proper substitution through the function predicate with a domain. (Contributed by Alexander van der Vekens, 15-Jul-2018.)
Assertion
Ref Expression
sbcfng (𝑋 𝑉 → ([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 4848 . . . 4 (𝐹 Fn A ↔ (Fun 𝐹 dom 𝐹 = A))
21a1i 9 . . 3 (𝑋 𝑉 → (𝐹 Fn A ↔ (Fun 𝐹 dom 𝐹 = A)))
32sbcbidv 2811 . 2 (𝑋 𝑉 → ([𝑋 / x]𝐹 Fn A[𝑋 / x](Fun 𝐹 dom 𝐹 = A)))
4 sbcfung 4868 . . . 4 (𝑋 𝑉 → ([𝑋 / x]Fun 𝐹 ↔ Fun 𝑋 / x𝐹))
5 sbceqg 2860 . . . . 5 (𝑋 𝑉 → ([𝑋 / x]dom 𝐹 = A𝑋 / xdom 𝐹 = 𝑋 / xA))
6 csbdmg 4472 . . . . . 6 (𝑋 𝑉𝑋 / xdom 𝐹 = dom 𝑋 / x𝐹)
76eqeq1d 2045 . . . . 5 (𝑋 𝑉 → (𝑋 / xdom 𝐹 = 𝑋 / xA ↔ dom 𝑋 / x𝐹 = 𝑋 / xA))
85, 7bitrd 177 . . . 4 (𝑋 𝑉 → ([𝑋 / x]dom 𝐹 = A ↔ dom 𝑋 / x𝐹 = 𝑋 / xA))
94, 8anbi12d 442 . . 3 (𝑋 𝑉 → (([𝑋 / x]Fun 𝐹 [𝑋 / x]dom 𝐹 = A) ↔ (Fun 𝑋 / x𝐹 dom 𝑋 / x𝐹 = 𝑋 / xA)))
10 sbcan 2799 . . 3 ([𝑋 / x](Fun 𝐹 dom 𝐹 = A) ↔ ([𝑋 / x]Fun 𝐹 [𝑋 / x]dom 𝐹 = A))
11 df-fn 4848 . . 3 (𝑋 / x𝐹 Fn 𝑋 / xA ↔ (Fun 𝑋 / x𝐹 dom 𝑋 / x𝐹 = 𝑋 / xA))
129, 10, 113bitr4g 212 . 2 (𝑋 𝑉 → ([𝑋 / x](Fun 𝐹 dom 𝐹 = A) ↔ 𝑋 / x𝐹 Fn 𝑋 / xA))
133, 12bitrd 177 1 (𝑋 𝑉 → ([𝑋 / x]𝐹 Fn A𝑋 / x𝐹 Fn 𝑋 / xA))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wb 98   = wceq 1242   wcel 1390  [wsbc 2758  csb 2846  dom cdm 4288  Fun wfun 4839   Fn wfn 4840
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-v 2553  df-sbc 2759  df-csb 2847  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-br 3756  df-opab 3810  df-id 4021  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-fun 4847  df-fn 4848
This theorem is referenced by:  sbcfg  4988
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