Theorem inres 5954

Description: Move intersection into class restriction. (Contributed by NM, 18-Dec-2008.)

Assertion

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

Proof of Theorem inres

Step	Hyp	Ref	Expression
1		inass 4178	. 2 ⊢ ((𝐴 ∩ 𝐵) ∩ (𝐶 × V)) = (𝐴 ∩ (𝐵 ∩ (𝐶 × V)))
2		df-res 5634	. 2 ⊢ ((𝐴 ∩ 𝐵) ↾ 𝐶) = ((𝐴 ∩ 𝐵) ∩ (𝐶 × V))
3		df-res 5634	. . 3 ⊢ (𝐵 ↾ 𝐶) = (𝐵 ∩ (𝐶 × V))
4	3	ineq2i 4167	. 2 ⊢ (𝐴 ∩ (𝐵 ↾ 𝐶)) = (𝐴 ∩ (𝐵 ∩ (𝐶 × V)))
5	1, 2, 4	3eqtr4ri 2768	1 ⊢ (𝐴 ∩ (𝐵 ↾ 𝐶)) = ((𝐴 ∩ 𝐵) ↾ 𝐶)

Syntax hints:   = wceq 1541  Vcvv 3438   ∩ cin 3898   × cxp 5620   ↾ cres 5624

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-ext 2706

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1544  df-ex 1781  df-sb 2068  df-clab 2713  df-cleq 2726  df-clel 2809  df-rab 3398  df-v 3440  df-in 3906  df-res 5634

This theorem is referenced by:  resindm  5987  fninfp  7118  symgcom2  33115  inres2  38382  xrnres2  38550  br1cossinres  38649