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 A ↔ ⦋𝑋 / x⦌𝐹 Fn ⦋𝑋 / x⦌A))

Distinct variable groups:   x,𝑉   x,𝑋

Allowed substitution hints:   A(x)   𝐹(x)

Proof of Theorem sbcfng

Step	Hyp	Ref	Expression
1		df-fn 4848	. . . 4 ⊢ (𝐹 Fn A ↔ (Fun 𝐹 ∧ dom 𝐹 = A))
2	1	a1i 9	. . 3 ⊢ (𝑋 ∈ 𝑉 → (𝐹 Fn A ↔ (Fun 𝐹 ∧ dom 𝐹 = A)))
3	2	sbcbidv 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 / x⦌𝐹))
5		sbceqg 2860	. . . . 5 ⊢ (𝑋 ∈ 𝑉 → ([𝑋 / x]dom 𝐹 = A ↔ ⦋𝑋 / x⦌dom 𝐹 = ⦋𝑋 / x⦌A))
6		csbdmg 4472	. . . . . 6 ⊢ (𝑋 ∈ 𝑉 → ⦋𝑋 / x⦌dom 𝐹 = dom ⦋𝑋 / x⦌𝐹)
7	6	eqeq1d 2045	. . . . 5 ⊢ (𝑋 ∈ 𝑉 → (⦋𝑋 / x⦌dom 𝐹 = ⦋𝑋 / x⦌A ↔ dom ⦋𝑋 / x⦌𝐹 = ⦋𝑋 / x⦌A))
8	5, 7	bitrd 177	. . . 4 ⊢ (𝑋 ∈ 𝑉 → ([𝑋 / x]dom 𝐹 = A ↔ dom ⦋𝑋 / x⦌𝐹 = ⦋𝑋 / x⦌A))
9	4, 8	anbi12d 442	. . 3 ⊢ (𝑋 ∈ 𝑉 → (([𝑋 / x]Fun 𝐹 ∧ [𝑋 / x]dom 𝐹 = A) ↔ (Fun ⦋𝑋 / x⦌𝐹 ∧ dom ⦋𝑋 / x⦌𝐹 = ⦋𝑋 / x⦌A)))
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 / x⦌𝐹 Fn ⦋𝑋 / x⦌A ↔ (Fun ⦋𝑋 / x⦌𝐹 ∧ dom ⦋𝑋 / x⦌𝐹 = ⦋𝑋 / x⦌A))
12	9, 10, 11	3bitr4g 212	. 2 ⊢ (𝑋 ∈ 𝑉 → ([𝑋 / x](Fun 𝐹 ∧ dom 𝐹 = A) ↔ ⦋𝑋 / x⦌𝐹 Fn ⦋𝑋 / x⦌A))
13	3, 12	bitrd 177	1 ⊢ (𝑋 ∈ 𝑉 → ([𝑋 / x]𝐹 Fn A ↔ ⦋𝑋 / x⦌𝐹 Fn ⦋𝑋 / x⦌A))

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