Theorem inres 4926

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 3347	. 2 ⊢ ((𝐴 ∩ 𝐵) ∩ (𝐶 × V)) = (𝐴 ∩ (𝐵 ∩ (𝐶 × V)))
2		df-res 4640	. 2 ⊢ ((𝐴 ∩ 𝐵) ↾ 𝐶) = ((𝐴 ∩ 𝐵) ∩ (𝐶 × V))
3		df-res 4640	. . 3 ⊢ (𝐵 ↾ 𝐶) = (𝐵 ∩ (𝐶 × V))
4	3	ineq2i 3335	. 2 ⊢ (𝐴 ∩ (𝐵 ↾ 𝐶)) = (𝐴 ∩ (𝐵 ∩ (𝐶 × V)))
5	1, 2, 4	3eqtr4ri 2209	1 ⊢ (𝐴 ∩ (𝐵 ↾ 𝐶)) = ((𝐴 ∩ 𝐵) ↾ 𝐶)

Syntax hints:   = wceq 1353  Vcvv 2739   ∩ cin 3130   × cxp 4626   ↾ cres 4630

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-v 2741  df-in 3137  df-res 4640

This theorem is referenced by:  resindm  4951