Theorem resindir 4646

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 3184	. 2 ⊢ ((𝐴 ∩ 𝐵) ∩ (𝐶 × V)) = ((𝐴 ∩ (𝐶 × V)) ∩ (𝐵 ∩ (𝐶 × V)))
2		df-res 4375	. 2 ⊢ ((𝐴 ∩ 𝐵) ↾ 𝐶) = ((𝐴 ∩ 𝐵) ∩ (𝐶 × V))
3		df-res 4375	. . 3 ⊢ (𝐴 ↾ 𝐶) = (𝐴 ∩ (𝐶 × V))
4		df-res 4375	. . 3 ⊢ (𝐵 ↾ 𝐶) = (𝐵 ∩ (𝐶 × V))
5	3, 4	ineq12i 3165	. 2 ⊢ ((𝐴 ↾ 𝐶) ∩ (𝐵 ↾ 𝐶)) = ((𝐴 ∩ (𝐶 × V)) ∩ (𝐵 ∩ (𝐶 × V)))
6	1, 2, 5	3eqtr4i 2111	1 ⊢ ((𝐴 ∩ 𝐵) ↾ 𝐶) = ((𝐴 ↾ 𝐶) ∩ (𝐵 ↾ 𝐶))

Syntax hints:   = wceq 1284  Vcvv 2601   ∩ cin 2972   × cxp 4361   ↾ cres 4365

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-v 2603  df-in 2979  df-res 4375

This theorem is referenced by:  inimass  4760  fnreseql  5298