Theorem csbres 5956

Description: Distribute proper substitution through the restriction of a class. (Contributed by Alan Sare, 10-Nov-2012.) (Revised by NM, 23-Aug-2018.)

Assertion

Ref	Expression
csbres	⊢ ⦋𝐴 / 𝑥⦌(𝐵 ↾ 𝐶) = (⦋𝐴 / 𝑥⦌𝐵 ↾ ⦋𝐴 / 𝑥⦌𝐶)

Proof of Theorem csbres

Step	Hyp	Ref	Expression
1		df-res 5653	. . 3 ⊢ (𝐵 ↾ 𝐶) = (𝐵 ∩ (𝐶 × V))
2	1	csbeq2i 3873	. 2 ⊢ ⦋𝐴 / 𝑥⦌(𝐵 ↾ 𝐶) = ⦋𝐴 / 𝑥⦌(𝐵 ∩ (𝐶 × V))
3		csbxp 5741	. . . . . 6 ⊢ ⦋𝐴 / 𝑥⦌(𝐶 × V) = (⦋𝐴 / 𝑥⦌𝐶 × ⦋𝐴 / 𝑥⦌V)
4		csbconstg 3884	. . . . . . 7 ⊢ (𝐴 ∈ V → ⦋𝐴 / 𝑥⦌V = V)
5	4	xpeq2d 5671	. . . . . 6 ⊢ (𝐴 ∈ V → (⦋𝐴 / 𝑥⦌𝐶 × ⦋𝐴 / 𝑥⦌V) = (⦋𝐴 / 𝑥⦌𝐶 × V))
6	3, 5	eqtrid 2777	. . . . 5 ⊢ (𝐴 ∈ V → ⦋𝐴 / 𝑥⦌(𝐶 × V) = (⦋𝐴 / 𝑥⦌𝐶 × V))
7		0xp 5740	. . . . . . 7 ⊢ (∅ × V) = ∅
8	7	a1i 11	. . . . . 6 ⊢ (¬ 𝐴 ∈ V → (∅ × V) = ∅)
9		csbprc 4375	. . . . . . 7 ⊢ (¬ 𝐴 ∈ V → ⦋𝐴 / 𝑥⦌𝐶 = ∅)
10	9	xpeq1d 5670	. . . . . 6 ⊢ (¬ 𝐴 ∈ V → (⦋𝐴 / 𝑥⦌𝐶 × V) = (∅ × V))
11		csbprc 4375	. . . . . 6 ⊢ (¬ 𝐴 ∈ V → ⦋𝐴 / 𝑥⦌(𝐶 × V) = ∅)
12	8, 10, 11	3eqtr4rd 2776	. . . . 5 ⊢ (¬ 𝐴 ∈ V → ⦋𝐴 / 𝑥⦌(𝐶 × V) = (⦋𝐴 / 𝑥⦌𝐶 × V))
13	6, 12	pm2.61i 182	. . . 4 ⊢ ⦋𝐴 / 𝑥⦌(𝐶 × V) = (⦋𝐴 / 𝑥⦌𝐶 × V)
14	13	ineq2i 4183	. . 3 ⊢ (⦋𝐴 / 𝑥⦌𝐵 ∩ ⦋𝐴 / 𝑥⦌(𝐶 × V)) = (⦋𝐴 / 𝑥⦌𝐵 ∩ (⦋𝐴 / 𝑥⦌𝐶 × V))
15		csbin 4408	. . 3 ⊢ ⦋𝐴 / 𝑥⦌(𝐵 ∩ (𝐶 × V)) = (⦋𝐴 / 𝑥⦌𝐵 ∩ ⦋𝐴 / 𝑥⦌(𝐶 × V))
16		df-res 5653	. . 3 ⊢ (⦋𝐴 / 𝑥⦌𝐵 ↾ ⦋𝐴 / 𝑥⦌𝐶) = (⦋𝐴 / 𝑥⦌𝐵 ∩ (⦋𝐴 / 𝑥⦌𝐶 × V))
17	14, 15, 16	3eqtr4i 2763	. 2 ⊢ ⦋𝐴 / 𝑥⦌(𝐵 ∩ (𝐶 × V)) = (⦋𝐴 / 𝑥⦌𝐵 ↾ ⦋𝐴 / 𝑥⦌𝐶)
18	2, 17	eqtri 2753	1 ⊢ ⦋𝐴 / 𝑥⦌(𝐵 ↾ 𝐶) = (⦋𝐴 / 𝑥⦌𝐵 ↾ ⦋𝐴 / 𝑥⦌𝐶)

Syntax hints:  ¬ wn 3   = wceq 1540   ∈ wcel 2109  Vcvv 3450  ⦋csb 3865   ∩ cin 3916  ∅c0 4299   × cxp 5639   ↾ cres 5643

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-sep 5254  ax-nul 5264  ax-pr 5390

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-rab 3409  df-v 3452  df-sbc 3757  df-csb 3866  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-nul 4300  df-if 4492  df-sn 4593  df-pr 4595  df-op 4599  df-opab 5173  df-xp 5647  df-res 5653

This theorem is referenced by:  csbfrecsg  8266  csbima12gALTVD  44893