Theorem sbcfng 5408

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 5262	. . . 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 3048	. 2 ⊢ (𝑋 ∈ 𝑉 → ([𝑋 / 𝑥]𝐹 Fn 𝐴 ↔ [𝑋 / 𝑥](Fun 𝐹 ∧ dom 𝐹 = 𝐴)))
4		sbcfung 5283	. . . 4 ⊢ (𝑋 ∈ 𝑉 → ([𝑋 / 𝑥]Fun 𝐹 ↔ Fun ⦋𝑋 / 𝑥⦌𝐹))
5		sbceqg 3100	. . . . 5 ⊢ (𝑋 ∈ 𝑉 → ([𝑋 / 𝑥]dom 𝐹 = 𝐴 ↔ ⦋𝑋 / 𝑥⦌dom 𝐹 = ⦋𝑋 / 𝑥⦌𝐴))
6		csbdmg 4861	. . . . . 6 ⊢ (𝑋 ∈ 𝑉 → ⦋𝑋 / 𝑥⦌dom 𝐹 = dom ⦋𝑋 / 𝑥⦌𝐹)
7	6	eqeq1d 2205	. . . . 5 ⊢ (𝑋 ∈ 𝑉 → (⦋𝑋 / 𝑥⦌dom 𝐹 = ⦋𝑋 / 𝑥⦌𝐴 ↔ dom ⦋𝑋 / 𝑥⦌𝐹 = ⦋𝑋 / 𝑥⦌𝐴))
8	5, 7	bitrd 188	. . . 4 ⊢ (𝑋 ∈ 𝑉 → ([𝑋 / 𝑥]dom 𝐹 = 𝐴 ↔ dom ⦋𝑋 / 𝑥⦌𝐹 = ⦋𝑋 / 𝑥⦌𝐴))
9	4, 8	anbi12d 473	. . 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 3032	. . 3 ⊢ ([𝑋 / 𝑥](Fun 𝐹 ∧ dom 𝐹 = 𝐴) ↔ ([𝑋 / 𝑥]Fun 𝐹 ∧ [𝑋 / 𝑥]dom 𝐹 = 𝐴))
11		df-fn 5262	. . 3 ⊢ (⦋𝑋 / 𝑥⦌𝐹 Fn ⦋𝑋 / 𝑥⦌𝐴 ↔ (Fun ⦋𝑋 / 𝑥⦌𝐹 ∧ dom ⦋𝑋 / 𝑥⦌𝐹 = ⦋𝑋 / 𝑥⦌𝐴))
12	9, 10, 11	3bitr4g 223	. 2 ⊢ (𝑋 ∈ 𝑉 → ([𝑋 / 𝑥](Fun 𝐹 ∧ dom 𝐹 = 𝐴) ↔ ⦋𝑋 / 𝑥⦌𝐹 Fn ⦋𝑋 / 𝑥⦌𝐴))
13	3, 12	bitrd 188	1 ⊢ (𝑋 ∈ 𝑉 → ([𝑋 / 𝑥]𝐹 Fn 𝐴 ↔ ⦋𝑋 / 𝑥⦌𝐹 Fn ⦋𝑋 / 𝑥⦌𝐴))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1364   ∈ wcel 2167  [wsbc 2989  ⦋csb 3084  dom cdm 4664  Fun wfun 5253   Fn wfn 5254

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-14 2170  ax-ext 2178  ax-sep 4152  ax-pow 4208  ax-pr 4243

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-v 2765  df-sbc 2990  df-csb 3085  df-un 3161  df-in 3163  df-ss 3170  df-pw 3608  df-sn 3629  df-pr 3630  df-op 3632  df-br 4035  df-opab 4096  df-id 4329  df-rel 4671  df-cnv 4672  df-co 4673  df-dm 4674  df-fun 5261  df-fn 5262

This theorem is referenced by:  sbcfg  5409