Theorem resres 5276

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 5001	. 2 |- ( ( A |` B ) |` C ) = ( ( A |` B ) i^i ( C X. _V ) )
2		df-res 5001	. . 3 |- ( A |` B ) = ( A i^i ( B X. _V ) )
3	2	ineq1i 3681	. 2 |- ( ( A |` B ) i^i ( C X. _V ) ) = ( ( A i^i ( B X. _V ) ) i^i ( C X. _V ) )
4		xpindir 5127	. . . 4 |- ( ( B i^i C ) X. _V ) = ( ( B X. _V ) i^i ( C X. _V ) )
5	4	ineq2i 3682	. . 3 |- ( A i^i ( ( B i^i C ) X. _V ) ) = ( A i^i ( ( B X. _V ) i^i ( C X. _V ) ) )
6		df-res 5001	. . 3 |- ( A |` ( B i^i C ) ) = ( A i^i ( ( B i^i C ) X. _V ) )
7		inass 3693	. . 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 2483	. 2 |- ( ( A i^i ( B X. _V ) ) i^i ( C X. _V ) ) = ( A |` ( B i^i C ) )
9	1, 3, 8	3eqtri 2476	1 |- ( ( A |` B ) |` C ) = ( A |` ( B i^i C ) )

Syntax hints:   = wceq 1383   _Vcvv 3095   i^i cin 3460   X. cxp 4987   |` cres 4991

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-sep 4558  ax-nul 4566  ax-pr 4676

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-ral 2798  df-rex 2799  df-rab 2802  df-v 3097  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3771  df-if 3927  df-sn 4015  df-pr 4017  df-op 4021  df-opab 4496  df-xp 4995  df-rel 4996  df-res 5001

This theorem is referenced by:  rescom  5288  resabs1  5292  resima2  5297  resmpt3  5314  resdisj  5426  rescnvcnv  5460  fresin  5744  resdif  5826  curry1  6877  curry2  6880  pmresg  7448  gruima  9183  rlimres  13363  lo1res  13364  rlimresb  13370  lo1eq  13373  rlimeq  13374  fsets  14640  setsid  14655  sscres  15174  gsumzres  16893  gsumzresOLD  16897  txkgen  20131  tsmsresOLD  20623  tsmsres  20624  ressxms  21006  ressms  21007  dvres  22293  dvres3a  22296  cpnres  22318  dvmptres3  22337  rlimcnp2  23274  df1stres  27500  df2ndres  27501  indf1ofs  28017  wfrlem4  29322  frrlem4  29366  fouriersw  31968  fouriercn  31969