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Theorem resres 5378
Description: The restriction of a restriction. (Contributed by NM, 27-Mar-2008.)
Assertion
Ref Expression
resres ((𝐴𝐵) ↾ 𝐶) = (𝐴 ↾ (𝐵𝐶))

Proof of Theorem resres
StepHypRef Expression
1 df-res 5096 . 2 ((𝐴𝐵) ↾ 𝐶) = ((𝐴𝐵) ∩ (𝐶 × V))
2 df-res 5096 . . 3 (𝐴𝐵) = (𝐴 ∩ (𝐵 × V))
32ineq1i 3794 . 2 ((𝐴𝐵) ∩ (𝐶 × V)) = ((𝐴 ∩ (𝐵 × V)) ∩ (𝐶 × V))
4 xpindir 5226 . . . 4 ((𝐵𝐶) × V) = ((𝐵 × V) ∩ (𝐶 × V))
54ineq2i 3795 . . 3 (𝐴 ∩ ((𝐵𝐶) × V)) = (𝐴 ∩ ((𝐵 × V) ∩ (𝐶 × V)))
6 df-res 5096 . . 3 (𝐴 ↾ (𝐵𝐶)) = (𝐴 ∩ ((𝐵𝐶) × V))
7 inass 3807 . . 3 ((𝐴 ∩ (𝐵 × V)) ∩ (𝐶 × V)) = (𝐴 ∩ ((𝐵 × V) ∩ (𝐶 × V)))
85, 6, 73eqtr4ri 2654 . 2 ((𝐴 ∩ (𝐵 × V)) ∩ (𝐶 × V)) = (𝐴 ↾ (𝐵𝐶))
91, 3, 83eqtri 2647 1 ((𝐴𝐵) ↾ 𝐶) = (𝐴 ↾ (𝐵𝐶))
Colors of variables: wff setvar class
Syntax hints:   = wceq 1480  Vcvv 3190  cin 3559   × cxp 5082  cres 5086
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4751  ax-nul 4759  ax-pr 4877
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-rab 2917  df-v 3192  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3898  df-if 4065  df-sn 4156  df-pr 4158  df-op 4162  df-opab 4684  df-xp 5090  df-rel 5091  df-res 5096
This theorem is referenced by:  rescom  5392  resabs1  5396  resima2  5401  resima2OLD  5402  resmpt3  5419  resdisj  5532  rescnvcnv  5566  fresin  6040  resdif  6124  curry1  7229  curry2  7232  wfrlem4  7378  pmresg  7845  gruima  9584  rlimres  14239  lo1res  14240  rlimresb  14246  lo1eq  14249  rlimeq  14250  fsets  15831  setsid  15854  sscres  16423  gsumzres  18250  txkgen  21395  tsmsres  21887  ressxms  22270  ressms  22271  dvres  23615  dvres3a  23618  cpnres  23640  dvmptres3  23659  rlimcnp2  24627  df1stres  29365  df2ndres  29366  indf1ofs  29911  frrlem4  31537  dfrcl2  37486  relexpaddss  37530  limsupresuz  39371  fouriersw  39785  fouriercn  39786
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