Theorem resres 5944

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 5630	. 2 ⊢ ((𝐴 ↾ 𝐵) ↾ 𝐶) = ((𝐴 ↾ 𝐵) ∩ (𝐶 × V))
2		df-res 5630	. . 3 ⊢ (𝐴 ↾ 𝐵) = (𝐴 ∩ (𝐵 × V))
3	2	ineq1i 4145	. 2 ⊢ ((𝐴 ↾ 𝐵) ∩ (𝐶 × V)) = ((𝐴 ∩ (𝐵 × V)) ∩ (𝐶 × V))
4		xpindir 5776	. . . 4 ⊢ ((𝐵 ∩ 𝐶) × V) = ((𝐵 × V) ∩ (𝐶 × V))
5	4	ineq2i 4146	. . 3 ⊢ (𝐴 ∩ ((𝐵 ∩ 𝐶) × V)) = (𝐴 ∩ ((𝐵 × V) ∩ (𝐶 × V)))
6		df-res 5630	. . 3 ⊢ (𝐴 ↾ (𝐵 ∩ 𝐶)) = (𝐴 ∩ ((𝐵 ∩ 𝐶) × V))
7		inass 4156	. . 3 ⊢ ((𝐴 ∩ (𝐵 × V)) ∩ (𝐶 × V)) = (𝐴 ∩ ((𝐵 × V) ∩ (𝐶 × V)))
8	5, 6, 7	3eqtr4ri 2773	. 2 ⊢ ((𝐴 ∩ (𝐵 × V)) ∩ (𝐶 × V)) = (𝐴 ↾ (𝐵 ∩ 𝐶))
9	1, 3, 8	3eqtri 2766	1 ⊢ ((𝐴 ↾ 𝐵) ↾ 𝐶) = (𝐴 ↾ (𝐵 ∩ 𝐶))

Syntax hints:   = wceq 1547  Vcvv 3431   ∩ cin 3882   × cxp 5616   ↾ cres 5620

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-ext 2711  ax-sep 5218  ax-pr 5362

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-sb 2074  df-clab 2718  df-cleq 2731  df-clel 2814  df-ral 3054  df-rex 3064  df-rab 3392  df-v 3433  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4262  df-if 4455  df-sn 4556  df-pr 4558  df-op 4562  df-opab 5135  df-xp 5624  df-rel 5625  df-res 5630

This theorem is referenced by:  rescom  5954  resabs1  5958  resima2  5968  resmpt3  5990  resdisj  6120  rescnvcnv  6155  fresin  6696  resdif  6788  curry1  8043  curry2  8046  frrlem4  8229  pmresg  8808  gruima  10716  rlimres  15511  lo1res  15512  rlimresb  15518  lo1eq  15521  rlimeq  15522  fsets  17130  setsid  17168  sscres  17781  gsumzres  19875  txkgen  23635  tsmsres  24127  ressxms  24508  ressms  24509  dvres  25896  dvres3a  25899  cpnres  25922  dvmptres3  25941  rlimcnp2  26948  df1stres  32796  df2ndres  32797  indf1ofs  32945  dfrcl2  44118  relexpaddss  44162  limsupresuz  46146  liminfresuz  46227  fouriersw  46674  fouriercn  46675  tposresg  49368  tposres3  49371