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Theorem sbcfng 6655

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 6491	. . . 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 3793	. 2 ⊢ (𝑋 ∈ 𝑉 → ([𝑋 / 𝑥]𝐹 Fn 𝐴 ↔ [𝑋 / 𝑥](Fun 𝐹 ∧ dom 𝐹 = 𝐴)))
4		sbcfung 6512	. . . 4 ⊢ (𝑋 ∈ 𝑉 → ([𝑋 / 𝑥]Fun 𝐹 ↔ Fun ⦋𝑋 / 𝑥⦌𝐹))
5		sbceqg 4361	. . . . 5 ⊢ (𝑋 ∈ 𝑉 → ([𝑋 / 𝑥]dom 𝐹 = 𝐴 ↔ ⦋𝑋 / 𝑥⦌dom 𝐹 = ⦋𝑋 / 𝑥⦌𝐴))
6		csbdm 5843	. . . . . 6 ⊢ ⦋𝑋 / 𝑥⦌dom 𝐹 = dom ⦋𝑋 / 𝑥⦌𝐹
7	6	eqeq1i 2738	. . . . 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 632	. . 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 3787	. . 3 ⊢ ([𝑋 / 𝑥](Fun 𝐹 ∧ dom 𝐹 = 𝐴) ↔ ([𝑋 / 𝑥]Fun 𝐹 ∧ [𝑋 / 𝑥]dom 𝐹 = 𝐴))
11		df-fn 6491	. . 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 206 ∧ wa 395 = wceq 1541 ∈ wcel 2113 [wsbc 3737 ⦋csb 3846 dom cdm 5621 Fun wfun 6482 Fn wfn 6483

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 1911 ax-6 1968 ax-7 2009 ax-8 2115 ax-9 2123 ax-10 2146 ax-11 2162 ax-12 2182 ax-ext 2705 ax-sep 5238 ax-nul 5248 ax-pr 5374

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2537 df-clab 2712 df-cleq 2725 df-clel 2808 df-nfc 2882 df-ral 3049 df-rex 3058 df-rab 3397 df-v 3439 df-sbc 3738 df-csb 3847 df-dif 3901 df-un 3903 df-ss 3915 df-nul 4283 df-if 4477 df-sn 4578 df-pr 4580 df-op 4584 df-br 5096 df-opab 5158 df-id 5516 df-xp 5627 df-rel 5628 df-cnv 5629 df-co 5630 df-dm 5631 df-fun 6490 df-fn 6491

This theorem is referenced by: sbcfg 6656

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