Theorem sbcfg 5956

Description: Distribute proper substitution through the function predicate with domain and codomain. (Contributed by Alexander van der Vekens, 15-Jul-2018.)

Assertion

Ref	Expression
sbcfg	⊢ (𝑋 ∈ 𝑉 → ([𝑋 / 𝑥]𝐹:𝐴⟶𝐵 ↔ ⦋𝑋 / 𝑥⦌𝐹:⦋𝑋 / 𝑥⦌𝐴⟶⦋𝑋 / 𝑥⦌𝐵))

Distinct variable groups:   𝑥,𝑉   𝑥,𝑋

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

Proof of Theorem sbcfg

Step	Hyp	Ref	Expression
1		df-f 5808	. . . 4 ⊢ (𝐹:𝐴⟶𝐵 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹 ⊆ 𝐵))
2	1	a1i 11	. . 3 ⊢ (𝑋 ∈ 𝑉 → (𝐹:𝐴⟶𝐵 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹 ⊆ 𝐵)))
3	2	sbcbidv 3457	. 2 ⊢ (𝑋 ∈ 𝑉 → ([𝑋 / 𝑥]𝐹:𝐴⟶𝐵 ↔ [𝑋 / 𝑥](𝐹 Fn 𝐴 ∧ ran 𝐹 ⊆ 𝐵)))
4		sbcfng 5955	. . . 4 ⊢ (𝑋 ∈ 𝑉 → ([𝑋 / 𝑥]𝐹 Fn 𝐴 ↔ ⦋𝑋 / 𝑥⦌𝐹 Fn ⦋𝑋 / 𝑥⦌𝐴))
5		sbcssg 4035	. . . . 5 ⊢ (𝑋 ∈ 𝑉 → ([𝑋 / 𝑥]ran 𝐹 ⊆ 𝐵 ↔ ⦋𝑋 / 𝑥⦌ran 𝐹 ⊆ ⦋𝑋 / 𝑥⦌𝐵))
6		csbrn 5514	. . . . . 6 ⊢ ⦋𝑋 / 𝑥⦌ran 𝐹 = ran ⦋𝑋 / 𝑥⦌𝐹
7	6	sseq1i 3592	. . . . 5 ⊢ (⦋𝑋 / 𝑥⦌ran 𝐹 ⊆ ⦋𝑋 / 𝑥⦌𝐵 ↔ ran ⦋𝑋 / 𝑥⦌𝐹 ⊆ ⦋𝑋 / 𝑥⦌𝐵)
8	5, 7	syl6bb 275	. . . 4 ⊢ (𝑋 ∈ 𝑉 → ([𝑋 / 𝑥]ran 𝐹 ⊆ 𝐵 ↔ ran ⦋𝑋 / 𝑥⦌𝐹 ⊆ ⦋𝑋 / 𝑥⦌𝐵))
9	4, 8	anbi12d 743	. . 3 ⊢ (𝑋 ∈ 𝑉 → (([𝑋 / 𝑥]𝐹 Fn 𝐴 ∧ [𝑋 / 𝑥]ran 𝐹 ⊆ 𝐵) ↔ (⦋𝑋 / 𝑥⦌𝐹 Fn ⦋𝑋 / 𝑥⦌𝐴 ∧ ran ⦋𝑋 / 𝑥⦌𝐹 ⊆ ⦋𝑋 / 𝑥⦌𝐵)))
10		sbcan 3445	. . 3 ⊢ ([𝑋 / 𝑥](𝐹 Fn 𝐴 ∧ ran 𝐹 ⊆ 𝐵) ↔ ([𝑋 / 𝑥]𝐹 Fn 𝐴 ∧ [𝑋 / 𝑥]ran 𝐹 ⊆ 𝐵))
11		df-f 5808	. . 3 ⊢ (⦋𝑋 / 𝑥⦌𝐹:⦋𝑋 / 𝑥⦌𝐴⟶⦋𝑋 / 𝑥⦌𝐵 ↔ (⦋𝑋 / 𝑥⦌𝐹 Fn ⦋𝑋 / 𝑥⦌𝐴 ∧ ran ⦋𝑋 / 𝑥⦌𝐹 ⊆ ⦋𝑋 / 𝑥⦌𝐵))
12	9, 10, 11	3bitr4g 302	. 2 ⊢ (𝑋 ∈ 𝑉 → ([𝑋 / 𝑥](𝐹 Fn 𝐴 ∧ ran 𝐹 ⊆ 𝐵) ↔ ⦋𝑋 / 𝑥⦌𝐹:⦋𝑋 / 𝑥⦌𝐴⟶⦋𝑋 / 𝑥⦌𝐵))
13	3, 12	bitrd 267	1 ⊢ (𝑋 ∈ 𝑉 → ([𝑋 / 𝑥]𝐹:𝐴⟶𝐵 ↔ ⦋𝑋 / 𝑥⦌𝐹:⦋𝑋 / 𝑥⦌𝐴⟶⦋𝑋 / 𝑥⦌𝐵))

Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   ∈ wcel 1977  [wsbc 3402  ⦋csb 3499   ⊆ wss 3540  ran crn 5039   Fn wfn 5799  ⟶wf 5800

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-fal 1481  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-br 4584  df-opab 4644  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-fun 5806  df-fn 5807  df-f 5808

This theorem is referenced by:  csbwrdg  13189