Theorem inres 5968

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 4191	. 2 ⊢ ((𝐴 ∩ 𝐵) ∩ (𝐶 × V)) = (𝐴 ∩ (𝐵 ∩ (𝐶 × V)))
2		df-res 5650	. 2 ⊢ ((𝐴 ∩ 𝐵) ↾ 𝐶) = ((𝐴 ∩ 𝐵) ∩ (𝐶 × V))
3		df-res 5650	. . 3 ⊢ (𝐵 ↾ 𝐶) = (𝐵 ∩ (𝐶 × V))
4	3	ineq2i 4180	. 2 ⊢ (𝐴 ∩ (𝐵 ↾ 𝐶)) = (𝐴 ∩ (𝐵 ∩ (𝐶 × V)))
5	1, 2, 4	3eqtr4ri 2763	1 ⊢ (𝐴 ∩ (𝐵 ↾ 𝐶)) = ((𝐴 ∩ 𝐵) ↾ 𝐶)

Syntax hints:   = wceq 1540  Vcvv 3447   ∩ cin 3913   × cxp 5636   ↾ cres 5640

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2701

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-rab 3406  df-v 3449  df-in 3921  df-res 5650

This theorem is referenced by:  resindm  6001  fninfp  7148  symgcom2  33041  inres2  38234  xrnres2  38389  br1cossinres  38438