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Theorem sbcfng 5277
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 5133 . . . 4 (𝐹 Fn 𝐴 ↔ (Fun 𝐹 ∧ dom 𝐹 = 𝐴))
21a1i 9 . . 3 (𝑋𝑉 → (𝐹 Fn 𝐴 ↔ (Fun 𝐹 ∧ dom 𝐹 = 𝐴)))
32sbcbidv 2970 . 2 (𝑋𝑉 → ([𝑋 / 𝑥]𝐹 Fn 𝐴[𝑋 / 𝑥](Fun 𝐹 ∧ dom 𝐹 = 𝐴)))
4 sbcfung 5154 . . . 4 (𝑋𝑉 → ([𝑋 / 𝑥]Fun 𝐹 ↔ Fun 𝑋 / 𝑥𝐹))
5 sbceqg 3022 . . . . 5 (𝑋𝑉 → ([𝑋 / 𝑥]dom 𝐹 = 𝐴𝑋 / 𝑥dom 𝐹 = 𝑋 / 𝑥𝐴))
6 csbdmg 4740 . . . . . 6 (𝑋𝑉𝑋 / 𝑥dom 𝐹 = dom 𝑋 / 𝑥𝐹)
76eqeq1d 2149 . . . . 5 (𝑋𝑉 → (𝑋 / 𝑥dom 𝐹 = 𝑋 / 𝑥𝐴 ↔ dom 𝑋 / 𝑥𝐹 = 𝑋 / 𝑥𝐴))
85, 7bitrd 187 . . . 4 (𝑋𝑉 → ([𝑋 / 𝑥]dom 𝐹 = 𝐴 ↔ dom 𝑋 / 𝑥𝐹 = 𝑋 / 𝑥𝐴))
94, 8anbi12d 465 . . 3 (𝑋𝑉 → (([𝑋 / 𝑥]Fun 𝐹[𝑋 / 𝑥]dom 𝐹 = 𝐴) ↔ (Fun 𝑋 / 𝑥𝐹 ∧ dom 𝑋 / 𝑥𝐹 = 𝑋 / 𝑥𝐴)))
10 sbcan 2954 . . 3 ([𝑋 / 𝑥](Fun 𝐹 ∧ dom 𝐹 = 𝐴) ↔ ([𝑋 / 𝑥]Fun 𝐹[𝑋 / 𝑥]dom 𝐹 = 𝐴))
11 df-fn 5133 . . 3 (𝑋 / 𝑥𝐹 Fn 𝑋 / 𝑥𝐴 ↔ (Fun 𝑋 / 𝑥𝐹 ∧ dom 𝑋 / 𝑥𝐹 = 𝑋 / 𝑥𝐴))
129, 10, 113bitr4g 222 . 2 (𝑋𝑉 → ([𝑋 / 𝑥](Fun 𝐹 ∧ dom 𝐹 = 𝐴) ↔ 𝑋 / 𝑥𝐹 Fn 𝑋 / 𝑥𝐴))
133, 12bitrd 187 1 (𝑋𝑉 → ([𝑋 / 𝑥]𝐹 Fn 𝐴𝑋 / 𝑥𝐹 Fn 𝑋 / 𝑥𝐴))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wb 104   = wceq 1332  wcel 1481  [wsbc 2912  csb 3006  dom cdm 4546  Fun wfun 5124   Fn wfn 5125
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4053  ax-pow 4105  ax-pr 4138
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-v 2691  df-sbc 2913  df-csb 3007  df-un 3079  df-in 3081  df-ss 3088  df-pw 3516  df-sn 3537  df-pr 3538  df-op 3540  df-br 3937  df-opab 3997  df-id 4222  df-rel 4553  df-cnv 4554  df-co 4555  df-dm 4556  df-fun 5132  df-fn 5133
This theorem is referenced by:  sbcfg  5278
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