Theorem sbcfg 6543

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 6384	. . . 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 3753	. 2 ⊢ (𝑋 ∈ 𝑉 → ([𝑋 / 𝑥]𝐹:𝐴⟶𝐵 ↔ [𝑋 / 𝑥](𝐹 Fn 𝐴 ∧ ran 𝐹 ⊆ 𝐵)))
4		sbcfng 6542	. . . 4 ⊢ (𝑋 ∈ 𝑉 → ([𝑋 / 𝑥]𝐹 Fn 𝐴 ↔ ⦋𝑋 / 𝑥⦌𝐹 Fn ⦋𝑋 / 𝑥⦌𝐴))
5		sbcssg 4435	. . . . 5 ⊢ (𝑋 ∈ 𝑉 → ([𝑋 / 𝑥]ran 𝐹 ⊆ 𝐵 ↔ ⦋𝑋 / 𝑥⦌ran 𝐹 ⊆ ⦋𝑋 / 𝑥⦌𝐵))
6		csbrn 6066	. . . . . 6 ⊢ ⦋𝑋 / 𝑥⦌ran 𝐹 = ran ⦋𝑋 / 𝑥⦌𝐹
7	6	sseq1i 3929	. . . . 5 ⊢ (⦋𝑋 / 𝑥⦌ran 𝐹 ⊆ ⦋𝑋 / 𝑥⦌𝐵 ↔ ran ⦋𝑋 / 𝑥⦌𝐹 ⊆ ⦋𝑋 / 𝑥⦌𝐵)
8	5, 7	bitrdi 290	. . . 4 ⊢ (𝑋 ∈ 𝑉 → ([𝑋 / 𝑥]ran 𝐹 ⊆ 𝐵 ↔ ran ⦋𝑋 / 𝑥⦌𝐹 ⊆ ⦋𝑋 / 𝑥⦌𝐵))
9	4, 8	anbi12d 634	. . 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 3746	. . 3 ⊢ ([𝑋 / 𝑥](𝐹 Fn 𝐴 ∧ ran 𝐹 ⊆ 𝐵) ↔ ([𝑋 / 𝑥]𝐹 Fn 𝐴 ∧ [𝑋 / 𝑥]ran 𝐹 ⊆ 𝐵))
11		df-f 6384	. . 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 317	. 2 ⊢ (𝑋 ∈ 𝑉 → ([𝑋 / 𝑥](𝐹 Fn 𝐴 ∧ ran 𝐹 ⊆ 𝐵) ↔ ⦋𝑋 / 𝑥⦌𝐹:⦋𝑋 / 𝑥⦌𝐴⟶⦋𝑋 / 𝑥⦌𝐵))
13	3, 12	bitrd 282	1 ⊢ (𝑋 ∈ 𝑉 → ([𝑋 / 𝑥]𝐹:𝐴⟶𝐵 ↔ ⦋𝑋 / 𝑥⦌𝐹:⦋𝑋 / 𝑥⦌𝐴⟶⦋𝑋 / 𝑥⦌𝐵))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   ∈ wcel 2110  [wsbc 3694  ⦋csb 3811   ⊆ wss 3866  ran crn 5552   Fn wfn 6375  ⟶wf 6376

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708  ax-sep 5192  ax-nul 5199  ax-pr 5322

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3410  df-sbc 3695  df-csb 3812  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-nul 4238  df-if 4440  df-sn 4542  df-pr 4544  df-op 4548  df-br 5054  df-opab 5116  df-id 5455  df-xp 5557  df-rel 5558  df-cnv 5559  df-co 5560  df-dm 5561  df-rn 5562  df-res 5563  df-ima 5564  df-fun 6382  df-fn 6383  df-f 6384

This theorem is referenced by:  csbwrdg  14099