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Theorem resres 5995
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 5689 . 2 ((𝐴𝐵) ↾ 𝐶) = ((𝐴𝐵) ∩ (𝐶 × V))
2 df-res 5689 . . 3 (𝐴𝐵) = (𝐴 ∩ (𝐵 × V))
32ineq1i 4209 . 2 ((𝐴𝐵) ∩ (𝐶 × V)) = ((𝐴 ∩ (𝐵 × V)) ∩ (𝐶 × V))
4 xpindir 5835 . . . 4 ((𝐵𝐶) × V) = ((𝐵 × V) ∩ (𝐶 × V))
54ineq2i 4210 . . 3 (𝐴 ∩ ((𝐵𝐶) × V)) = (𝐴 ∩ ((𝐵 × V) ∩ (𝐶 × V)))
6 df-res 5689 . . 3 (𝐴 ↾ (𝐵𝐶)) = (𝐴 ∩ ((𝐵𝐶) × V))
7 inass 4220 . . 3 ((𝐴 ∩ (𝐵 × V)) ∩ (𝐶 × V)) = (𝐴 ∩ ((𝐵 × V) ∩ (𝐶 × V)))
85, 6, 73eqtr4ri 2772 . 2 ((𝐴 ∩ (𝐵 × V)) ∩ (𝐶 × V)) = (𝐴 ↾ (𝐵𝐶))
91, 3, 83eqtri 2765 1 ((𝐴𝐵) ↾ 𝐶) = (𝐴 ↾ (𝐵𝐶))
Colors of variables: wff setvar class
Syntax hints:   = wceq 1542  Vcvv 3475  cin 3948   × cxp 5675  cres 5679
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pr 5428
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-opab 5212  df-xp 5683  df-rel 5684  df-res 5689
This theorem is referenced by:  rescom  6008  resabs1  6012  resima2  6017  resmpt3  6039  resdisj  6169  rescnvcnv  6204  fresin  6761  resdif  6855  curry1  8090  curry2  8093  frrlem4  8274  wfrlem4OLD  8312  pmresg  8864  gruima  10797  rlimres  15502  lo1res  15503  rlimresb  15509  lo1eq  15512  rlimeq  15513  fsets  17102  setsid  17141  sscres  17770  gsumzres  19777  txkgen  23156  tsmsres  23648  ressxms  24034  ressms  24035  dvres  25428  dvres3a  25431  cpnres  25454  dvmptres3  25473  rlimcnp2  26471  df1stres  31925  df2ndres  31926  indf1ofs  33024  dfrcl2  42425  relexpaddss  42469  limsupresuz  44419  liminfresuz  44500  fouriersw  44947  fouriercn  44948
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