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Theorem sbcfng 5072
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 4933 . . . 4 (𝐹 Fn 𝐴 ↔ (Fun 𝐹 ∧ dom 𝐹 = 𝐴))
21a1i 9 . . 3 (𝑋𝑉 → (𝐹 Fn 𝐴 ↔ (Fun 𝐹 ∧ dom 𝐹 = 𝐴)))
32sbcbidv 2844 . 2 (𝑋𝑉 → ([𝑋 / 𝑥]𝐹 Fn 𝐴[𝑋 / 𝑥](Fun 𝐹 ∧ dom 𝐹 = 𝐴)))
4 sbcfung 4953 . . . 4 (𝑋𝑉 → ([𝑋 / 𝑥]Fun 𝐹 ↔ Fun 𝑋 / 𝑥𝐹))
5 sbceqg 2894 . . . . 5 (𝑋𝑉 → ([𝑋 / 𝑥]dom 𝐹 = 𝐴𝑋 / 𝑥dom 𝐹 = 𝑋 / 𝑥𝐴))
6 csbdmg 4557 . . . . . 6 (𝑋𝑉𝑋 / 𝑥dom 𝐹 = dom 𝑋 / 𝑥𝐹)
76eqeq1d 2064 . . . . 5 (𝑋𝑉 → (𝑋 / 𝑥dom 𝐹 = 𝑋 / 𝑥𝐴 ↔ dom 𝑋 / 𝑥𝐹 = 𝑋 / 𝑥𝐴))
85, 7bitrd 181 . . . 4 (𝑋𝑉 → ([𝑋 / 𝑥]dom 𝐹 = 𝐴 ↔ dom 𝑋 / 𝑥𝐹 = 𝑋 / 𝑥𝐴))
94, 8anbi12d 450 . . 3 (𝑋𝑉 → (([𝑋 / 𝑥]Fun 𝐹[𝑋 / 𝑥]dom 𝐹 = 𝐴) ↔ (Fun 𝑋 / 𝑥𝐹 ∧ dom 𝑋 / 𝑥𝐹 = 𝑋 / 𝑥𝐴)))
10 sbcan 2828 . . 3 ([𝑋 / 𝑥](Fun 𝐹 ∧ dom 𝐹 = 𝐴) ↔ ([𝑋 / 𝑥]Fun 𝐹[𝑋 / 𝑥]dom 𝐹 = 𝐴))
11 df-fn 4933 . . 3 (𝑋 / 𝑥𝐹 Fn 𝑋 / 𝑥𝐴 ↔ (Fun 𝑋 / 𝑥𝐹 ∧ dom 𝑋 / 𝑥𝐹 = 𝑋 / 𝑥𝐴))
129, 10, 113bitr4g 216 . 2 (𝑋𝑉 → ([𝑋 / 𝑥](Fun 𝐹 ∧ dom 𝐹 = 𝐴) ↔ 𝑋 / 𝑥𝐹 Fn 𝑋 / 𝑥𝐴))
133, 12bitrd 181 1 (𝑋𝑉 → ([𝑋 / 𝑥]𝐹 Fn 𝐴𝑋 / 𝑥𝐹 Fn 𝑋 / 𝑥𝐴))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 101  wb 102   = wceq 1259  wcel 1409  [wsbc 2787  csb 2880  dom cdm 4373  Fun wfun 4924   Fn wfn 4925
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3903  ax-pow 3955  ax-pr 3972
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-v 2576  df-sbc 2788  df-csb 2881  df-un 2950  df-in 2952  df-ss 2959  df-pw 3389  df-sn 3409  df-pr 3410  df-op 3412  df-br 3793  df-opab 3847  df-id 4058  df-rel 4380  df-cnv 4381  df-co 4382  df-dm 4383  df-fun 4932  df-fn 4933
This theorem is referenced by:  sbcfg  5073
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