Theorem resindi 5332

Description: Class restriction distributes over intersection. (Contributed by FL, 6-Oct-2008.)

Assertion

Ref	Expression
resindi	⊢ (𝐴 ↾ (𝐵 ∩ 𝐶)) = ((𝐴 ↾ 𝐵) ∩ (𝐴 ↾ 𝐶))

Proof of Theorem resindi

Step	Hyp	Ref	Expression
1		xpindir 5178	. . . 4 ⊢ ((𝐵 ∩ 𝐶) × V) = ((𝐵 × V) ∩ (𝐶 × V))
2	1	ineq2i 3773	. . 3 ⊢ (𝐴 ∩ ((𝐵 ∩ 𝐶) × V)) = (𝐴 ∩ ((𝐵 × V) ∩ (𝐶 × V)))
3		inindi 3792	. . 3 ⊢ (𝐴 ∩ ((𝐵 × V) ∩ (𝐶 × V))) = ((𝐴 ∩ (𝐵 × V)) ∩ (𝐴 ∩ (𝐶 × V)))
4	2, 3	eqtri 2632	. 2 ⊢ (𝐴 ∩ ((𝐵 ∩ 𝐶) × V)) = ((𝐴 ∩ (𝐵 × V)) ∩ (𝐴 ∩ (𝐶 × V)))
5		df-res 5050	. 2 ⊢ (𝐴 ↾ (𝐵 ∩ 𝐶)) = (𝐴 ∩ ((𝐵 ∩ 𝐶) × V))
6		df-res 5050	. . 3 ⊢ (𝐴 ↾ 𝐵) = (𝐴 ∩ (𝐵 × V))
7		df-res 5050	. . 3 ⊢ (𝐴 ↾ 𝐶) = (𝐴 ∩ (𝐶 × V))
8	6, 7	ineq12i 3774	. 2 ⊢ ((𝐴 ↾ 𝐵) ∩ (𝐴 ↾ 𝐶)) = ((𝐴 ∩ (𝐵 × V)) ∩ (𝐴 ∩ (𝐶 × V)))
9	4, 5, 8	3eqtr4i 2642	1 ⊢ (𝐴 ↾ (𝐵 ∩ 𝐶)) = ((𝐴 ↾ 𝐵) ∩ (𝐴 ↾ 𝐶))

Syntax hints:   = wceq 1475  Vcvv 3173   ∩ cin 3539   × cxp 5036   ↾ cres 5040

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-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-rab 2905  df-v 3175  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-opab 4644  df-xp 5044  df-rel 5045  df-res 5050

This theorem is referenced by:  resindm  5364  gsum2dlem2  18193  fnresin  28812  resisresindm  40320