Theorem resres 5284

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

Assertion

Ref	Expression
resres	|- ( ( A |` B ) |` C ) = ( A |` ( B i^i C ) )

Proof of Theorem resres

Step	Hyp	Ref	Expression
1		df-res 5011	. 2 |- ( ( A |` B ) |` C ) = ( ( A |` B ) i^i ( C X. _V ) )
2		df-res 5011	. . 3 |- ( A |` B ) = ( A i^i ( B X. _V ) )
3	2	ineq1i 3696	. 2 |- ( ( A |` B ) i^i ( C X. _V ) ) = ( ( A i^i ( B X. _V ) ) i^i ( C X. _V ) )
4		xpindir 5135	. . . 4 |- ( ( B i^i C ) X. _V ) = ( ( B X. _V ) i^i ( C X. _V ) )
5	4	ineq2i 3697	. . 3 |- ( A i^i ( ( B i^i C ) X. _V ) ) = ( A i^i ( ( B X. _V ) i^i ( C X. _V ) ) )
6		df-res 5011	. . 3 |- ( A |` ( B i^i C ) ) = ( A i^i ( ( B i^i C ) X. _V ) )
7		inass 3708	. . 3 |- ( ( A i^i ( B X. _V ) ) i^i ( C X. _V ) ) = ( A i^i ( ( B X. _V ) i^i ( C X. _V ) ) )
8	5, 6, 7	3eqtr4ri 2507	. 2 |- ( ( A i^i ( B X. _V ) ) i^i ( C X. _V ) ) = ( A |` ( B i^i C ) )
9	1, 3, 8	3eqtri 2500	1 |- ( ( A |` B ) |` C ) = ( A |` ( B i^i C ) )

Syntax hints:   = wceq 1379   _Vcvv 3113   i^i cin 3475   X. cxp 4997   |` cres 5001

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pr 4686

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-opab 4506  df-xp 5005  df-rel 5006  df-res 5011

This theorem is referenced by:  rescom  5296  resabs1  5300  resima2  5305  resmpt3  5322  resdisj  5434  rescnvcnv  5468  fresin  5752  resdif  5834  curry1  6872  curry2  6875  pmresg  7443  gruima  9176  rlimres  13340  lo1res  13341  rlimresb  13347  lo1eq  13350  rlimeq  13351  fsets  14512  setsid  14527  sscres  15049  gsumzres  16705  gsumzresOLD  16709  txkgen  19888  tsmsresOLD  20380  tsmsres  20381  ressxms  20763  ressms  20764  dvres  22050  dvres3a  22053  cpnres  22075  dvmptres3  22094  rlimcnp2  23024  df1stres  27194  df2ndres  27195  indf1ofs  27679  wfrlem4  28923  frrlem4  28967  fouriersw  31532  fouriercn  31533