Theorem resindir 5994

Description: Class restriction distributes over intersection. (Contributed by NM, 18-Dec-2008.)

Assertion

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

Proof of Theorem resindir

Step	Hyp	Ref	Expression
1		inindir 4187	. 2 ⊢ ((𝐴 ∩ 𝐵) ∩ (𝐶 × V)) = ((𝐴 ∩ (𝐶 × V)) ∩ (𝐵 ∩ (𝐶 × V)))
2		df-res 5672	. 2 ⊢ ((𝐴 ∩ 𝐵) ↾ 𝐶) = ((𝐴 ∩ 𝐵) ∩ (𝐶 × V))
3		df-res 5672	. . 3 ⊢ (𝐴 ↾ 𝐶) = (𝐴 ∩ (𝐶 × V))
4		df-res 5672	. . 3 ⊢ (𝐵 ↾ 𝐶) = (𝐵 ∩ (𝐶 × V))
5	3, 4	ineq12i 4170	. 2 ⊢ ((𝐴 ↾ 𝐶) ∩ (𝐵 ↾ 𝐶)) = ((𝐴 ∩ (𝐶 × V)) ∩ (𝐵 ∩ (𝐶 × V)))
6	1, 2, 5	3eqtr4i 2795	1 ⊢ ((𝐴 ∩ 𝐵) ↾ 𝐶) = ((𝐴 ↾ 𝐶) ∩ (𝐵 ↾ 𝐶))

This proof depends on syntax axioms:   = wceq 1569  Vcvv 3454   ∩ cin 3903   × cxp 5658   ↾ cres 5662

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-8 2144  ax-9 2152  ax-ext 2734

This proof depends on definitions:  df-bi 210  df-an 401  df-tru 1572  df-ex 1809  df-sb 2096  df-clab 2741  df-cleq 2754  df-clel 2837  df-rab 3416  df-v 3456  df-in 3911  df-res 5672

This theorem is used by:  inimass  6151  fnreseql  7043  xrnres3  39104