Theorem inres 5990

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 4173	. 2 ⊢ ((𝐴 ∩ 𝐵) ∩ (𝐶 × V)) = (𝐴 ∩ (𝐵 ∩ (𝐶 × V)))
2		df-res 5667	. 2 ⊢ ((𝐴 ∩ 𝐵) ↾ 𝐶) = ((𝐴 ∩ 𝐵) ∩ (𝐶 × V))
3		df-res 5667	. . 3 ⊢ (𝐵 ↾ 𝐶) = (𝐵 ∩ (𝐶 × V))
4	3	ineq2i 4163	. 2 ⊢ (𝐴 ∩ (𝐵 ↾ 𝐶)) = (𝐴 ∩ (𝐵 ∩ (𝐶 × V)))
5	1, 2, 4	3eqtr4ri 2794	1 ⊢ (𝐴 ∩ (𝐵 ↾ 𝐶)) = ((𝐴 ∩ 𝐵) ↾ 𝐶)

This proof depends on syntax axioms:   = wceq 1570  Vcvv 3450   ∩ cin 3898   × cxp 5653   ↾ cres 5657

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-ex 1813  df-sb 2100  df-clab 2739  df-cleq 2752  df-clel 2835  df-rab 3413  df-v 3452  df-in 3906  df-res 5667

This theorem is used by:  resindmOLD  6024  fninfp  7173  symgcom2  33527  inres2  38998  xrnres2  39177  br1cossinres  39288