Theorem csbres 5972

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 5664	. . 3 ⊢ (𝐵 ↾ 𝐶) = (𝐵 ∩ (𝐶 × V))
2	1	csbeq2i 3863	. 2 ⊢ ⦋𝐴 / 𝑥⦌(𝐵 ↾ 𝐶) = ⦋𝐴 / 𝑥⦌(𝐵 ∩ (𝐶 × V))
3		csbxp 5753	. . . . . 6 ⊢ ⦋𝐴 / 𝑥⦌(𝐶 × V) = (⦋𝐴 / 𝑥⦌𝐶 × ⦋𝐴 / 𝑥⦌V)
4		csbconstg 3874	. . . . . . 7 ⊢ (𝐴 ∈ V → ⦋𝐴 / 𝑥⦌V = V)
5	4	xpeq2d 5682	. . . . . 6 ⊢ (𝐴 ∈ V → (⦋𝐴 / 𝑥⦌𝐶 × ⦋𝐴 / 𝑥⦌V) = (⦋𝐴 / 𝑥⦌𝐶 × V))
6	3, 5	eqtrid 2812	. . . . 5 ⊢ (𝐴 ∈ V → ⦋𝐴 / 𝑥⦌(𝐶 × V) = (⦋𝐴 / 𝑥⦌𝐶 × V))
7		0xp 5751	. . . . . . 7 ⊢ (∅ × V) = ∅
8	7	a1i 11	. . . . . 6 ⊢ (¬ 𝐴 ∈ V → (∅ × V) = ∅)
9		csbprc 4366	. . . . . . 7 ⊢ (¬ 𝐴 ∈ V → ⦋𝐴 / 𝑥⦌𝐶 = ∅)
10	9	xpeq1d 5681	. . . . . 6 ⊢ (¬ 𝐴 ∈ V → (⦋𝐴 / 𝑥⦌𝐶 × V) = (∅ × V))
11		csbprc 4366	. . . . . 6 ⊢ (¬ 𝐴 ∈ V → ⦋𝐴 / 𝑥⦌(𝐶 × V) = ∅)
12	8, 10, 11	3eqtr4rd 2811	. . . . 5 ⊢ (¬ 𝐴 ∈ V → ⦋𝐴 / 𝑥⦌(𝐶 × V) = (⦋𝐴 / 𝑥⦌𝐶 × V))
13	6, 12	pm2.61i 184	. . . 4 ⊢ ⦋𝐴 / 𝑥⦌(𝐶 × V) = (⦋𝐴 / 𝑥⦌𝐶 × V)
14	13	ineq2i 4172	. . 3 ⊢ (⦋𝐴 / 𝑥⦌𝐵 ∩ ⦋𝐴 / 𝑥⦌(𝐶 × V)) = (⦋𝐴 / 𝑥⦌𝐵 ∩ (⦋𝐴 / 𝑥⦌𝐶 × V))
15		csbin 4399	. . 3 ⊢ ⦋𝐴 / 𝑥⦌(𝐵 ∩ (𝐶 × V)) = (⦋𝐴 / 𝑥⦌𝐵 ∩ ⦋𝐴 / 𝑥⦌(𝐶 × V))
16		df-res 5664	. . 3 ⊢ (⦋𝐴 / 𝑥⦌𝐵 ↾ ⦋𝐴 / 𝑥⦌𝐶) = (⦋𝐴 / 𝑥⦌𝐵 ∩ (⦋𝐴 / 𝑥⦌𝐶 × V))
17	14, 15, 16	3eqtr4i 2798	. 2 ⊢ ⦋𝐴 / 𝑥⦌(𝐵 ∩ (𝐶 × V)) = (⦋𝐴 / 𝑥⦌𝐵 ↾ ⦋𝐴 / 𝑥⦌𝐶)
18	2, 17	eqtri 2788	1 ⊢ ⦋𝐴 / 𝑥⦌(𝐵 ↾ 𝐶) = (⦋𝐴 / 𝑥⦌𝐵 ↾ ⦋𝐴 / 𝑥⦌𝐶)

Syntax hints:  ¬ wn 3   = wceq 1563   ∈ wcel 2145  Vcvv 3457  ⦋csb 3855   ∩ cin 3906  ∅c0 4288   × cxp 5650   ↾ cres 5654

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2737

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-nf 1807  df-sb 2094  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-rab 3418  df-v 3459  df-sbc 3748  df-csb 3856  df-dif 3910  df-in 3914  df-nul 4289  df-opab 5168  df-xp 5658  df-res 5664

This theorem is referenced by:  csbfrecsg  8269  csbima12gALTVD  45470