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Theorem sbcfng 6714

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 6546	. . . 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 3836	. 2 ⊢ (𝑋 ∈ 𝑉 → ([𝑋 / 𝑥]𝐹 Fn 𝐴 ↔ [𝑋 / 𝑥](Fun 𝐹 ∧ dom 𝐹 = 𝐴)))
4		sbcfung 6572	. . . 4 ⊢ (𝑋 ∈ 𝑉 → ([𝑋 / 𝑥]Fun 𝐹 ↔ Fun ⦋𝑋 / 𝑥⦌𝐹))
5		sbceqg 4409	. . . . 5 ⊢ (𝑋 ∈ 𝑉 → ([𝑋 / 𝑥]dom 𝐹 = 𝐴 ↔ ⦋𝑋 / 𝑥⦌dom 𝐹 = ⦋𝑋 / 𝑥⦌𝐴))
6		csbdm 5897	. . . . . 6 ⊢ ⦋𝑋 / 𝑥⦌dom 𝐹 = dom ⦋𝑋 / 𝑥⦌𝐹
7	6	eqeq1i 2736	. . . . 5 ⊢ (⦋𝑋 / 𝑥⦌dom 𝐹 = ⦋𝑋 / 𝑥⦌𝐴 ↔ dom ⦋𝑋 / 𝑥⦌𝐹 = ⦋𝑋 / 𝑥⦌𝐴)
8	5, 7	bitrdi 287	. . . 4 ⊢ (𝑋 ∈ 𝑉 → ([𝑋 / 𝑥]dom 𝐹 = 𝐴 ↔ dom ⦋𝑋 / 𝑥⦌𝐹 = ⦋𝑋 / 𝑥⦌𝐴))
9	4, 8	anbi12d 630	. . 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 3829	. . 3 ⊢ ([𝑋 / 𝑥](Fun 𝐹 ∧ dom 𝐹 = 𝐴) ↔ ([𝑋 / 𝑥]Fun 𝐹 ∧ [𝑋 / 𝑥]dom 𝐹 = 𝐴))
11		df-fn 6546	. . 3 ⊢ (⦋𝑋 / 𝑥⦌𝐹 Fn ⦋𝑋 / 𝑥⦌𝐴 ↔ (Fun ⦋𝑋 / 𝑥⦌𝐹 ∧ dom ⦋𝑋 / 𝑥⦌𝐹 = ⦋𝑋 / 𝑥⦌𝐴))
12	9, 10, 11	3bitr4g 314	. 2 ⊢ (𝑋 ∈ 𝑉 → ([𝑋 / 𝑥](Fun 𝐹 ∧ dom 𝐹 = 𝐴) ↔ ⦋𝑋 / 𝑥⦌𝐹 Fn ⦋𝑋 / 𝑥⦌𝐴))
13	3, 12	bitrd 279	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 395 = wceq 1540 ∈ wcel 2105 [wsbc 3777 ⦋csb 3893 dom cdm 5676 Fun wfun 6537 Fn wfn 6538

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-sep 5299 ax-nul 5306 ax-pr 5427

This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ral 3061 df-rex 3070 df-rab 3432 df-v 3475 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-sn 4629 df-pr 4631 df-op 4635 df-br 5149 df-opab 5211 df-id 5574 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-fun 6545 df-fn 6546

This theorem is referenced by: sbcfg 6715

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