Theorem resres 4878

Description: The restriction of a restriction. (Contributed by NM, 27-Mar-2008.)

Assertion

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

Proof of Theorem resres

Step	Hyp	Ref	Expression
1		df-res 4598	. 2 ⊢ ((𝐴 ↾ 𝐵) ↾ 𝐶) = ((𝐴 ↾ 𝐵) ∩ (𝐶 × V))
2		df-res 4598	. . 3 ⊢ (𝐴 ↾ 𝐵) = (𝐴 ∩ (𝐵 × V))
3	2	ineq1i 3304	. 2 ⊢ ((𝐴 ↾ 𝐵) ∩ (𝐶 × V)) = ((𝐴 ∩ (𝐵 × V)) ∩ (𝐶 × V))
4		xpindir 4722	. . . 4 ⊢ ((𝐵 ∩ 𝐶) × V) = ((𝐵 × V) ∩ (𝐶 × V))
5	4	ineq2i 3305	. . 3 ⊢ (𝐴 ∩ ((𝐵 ∩ 𝐶) × V)) = (𝐴 ∩ ((𝐵 × V) ∩ (𝐶 × V)))
6		df-res 4598	. . 3 ⊢ (𝐴 ↾ (𝐵 ∩ 𝐶)) = (𝐴 ∩ ((𝐵 ∩ 𝐶) × V))
7		inass 3317	. . 3 ⊢ ((𝐴 ∩ (𝐵 × V)) ∩ (𝐶 × V)) = (𝐴 ∩ ((𝐵 × V) ∩ (𝐶 × V)))
8	5, 6, 7	3eqtr4ri 2189	. 2 ⊢ ((𝐴 ∩ (𝐵 × V)) ∩ (𝐶 × V)) = (𝐴 ↾ (𝐵 ∩ 𝐶))
9	1, 3, 8	3eqtri 2182	1 ⊢ ((𝐴 ↾ 𝐵) ↾ 𝐶) = (𝐴 ↾ (𝐵 ∩ 𝐶))

Syntax hints:   = wceq 1335  Vcvv 2712   ∩ cin 3101   × cxp 4584   ↾ cres 4588

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-14 2131  ax-ext 2139  ax-sep 4082  ax-pow 4135  ax-pr 4169

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ral 2440  df-rex 2441  df-v 2714  df-un 3106  df-in 3108  df-ss 3115  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-opab 4026  df-xp 4592  df-rel 4593  df-res 4598

This theorem is referenced by:  rescom  4891  resabs1  4895  resima2  4900  resmpt3  4915  resdisj  5014  rescnvcnv  5048  funimaexg  5254  fresin  5348  resdif  5436  pmresg  6621  setsslid  12251