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Theorem sbcfng 6711

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 6543	. . . 4 ⊢ (𝐹 Fn 𝐴 ↔ (Fun 𝐹 ∧ dom 𝐹 = 𝐴))
2	1	a1i 11	. . 3 ⊢ (𝑋 ∈ 𝑉 → (𝐹 Fn 𝐴 ↔ (Fun 𝐹 ∧ dom 𝐹 = 𝐴)))
3	2	sbcbidv 3835	. 2 ⊢ (𝑋 ∈ 𝑉 → ([𝑋 / 𝑥]𝐹 Fn 𝐴 ↔ [𝑋 / 𝑥](Fun 𝐹 ∧ dom 𝐹 = 𝐴)))
4		sbcfung 6569	. . . 4 ⊢ (𝑋 ∈ 𝑉 → ([𝑋 / 𝑥]Fun 𝐹 ↔ Fun ⦋𝑋 / 𝑥⦌𝐹))
5		sbceqg 4408	. . . . 5 ⊢ (𝑋 ∈ 𝑉 → ([𝑋 / 𝑥]dom 𝐹 = 𝐴 ↔ ⦋𝑋 / 𝑥⦌dom 𝐹 = ⦋𝑋 / 𝑥⦌𝐴))
6		csbdm 5895	. . . . . 6 ⊢ ⦋𝑋 / 𝑥⦌dom 𝐹 = dom ⦋𝑋 / 𝑥⦌𝐹
7	6	eqeq1i 2737	. . . . 5 ⊢ (⦋𝑋 / 𝑥⦌dom 𝐹 = ⦋𝑋 / 𝑥⦌𝐴 ↔ dom ⦋𝑋 / 𝑥⦌𝐹 = ⦋𝑋 / 𝑥⦌𝐴)
8	5, 7	bitrdi 286	. . . 4 ⊢ (𝑋 ∈ 𝑉 → ([𝑋 / 𝑥]dom 𝐹 = 𝐴 ↔ dom ⦋𝑋 / 𝑥⦌𝐹 = ⦋𝑋 / 𝑥⦌𝐴))
9	4, 8	anbi12d 631	. . 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 3828	. . 3 ⊢ ([𝑋 / 𝑥](Fun 𝐹 ∧ dom 𝐹 = 𝐴) ↔ ([𝑋 / 𝑥]Fun 𝐹 ∧ [𝑋 / 𝑥]dom 𝐹 = 𝐴))
11		df-fn 6543	. . 3 ⊢ (⦋𝑋 / 𝑥⦌𝐹 Fn ⦋𝑋 / 𝑥⦌𝐴 ↔ (Fun ⦋𝑋 / 𝑥⦌𝐹 ∧ dom ⦋𝑋 / 𝑥⦌𝐹 = ⦋𝑋 / 𝑥⦌𝐴))
12	9, 10, 11	3bitr4g 313	. 2 ⊢ (𝑋 ∈ 𝑉 → ([𝑋 / 𝑥](Fun 𝐹 ∧ dom 𝐹 = 𝐴) ↔ ⦋𝑋 / 𝑥⦌𝐹 Fn ⦋𝑋 / 𝑥⦌𝐴))
13	3, 12	bitrd 278	1 ⊢ (𝑋 ∈ 𝑉 → ([𝑋 / 𝑥]𝐹 Fn 𝐴 ↔ ⦋𝑋 / 𝑥⦌𝐹 Fn ⦋𝑋 / 𝑥⦌𝐴))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 = wceq 1541 ∈ wcel 2106 [wsbc 3776 ⦋csb 3892 dom cdm 5675 Fun wfun 6534 Fn wfn 6535

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-sep 5298 ax-nul 5305 ax-pr 5426

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ral 3062 df-rex 3071 df-rab 3433 df-v 3476 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4322 df-if 4528 df-sn 4628 df-pr 4630 df-op 4634 df-br 5148 df-opab 5210 df-id 5573 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-fun 6542 df-fn 6543

This theorem is referenced by: sbcfg 6712

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