Theorem resres 5025

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 4737	. 2 |- ( ( A |` B ) |` C ) = ( ( A |` B ) i^i ( C X. _V ) )
2		df-res 4737	. . 3 |- ( A |` B ) = ( A i^i ( B X. _V ) )
3	2	ineq1i 3404	. 2 |- ( ( A |` B ) i^i ( C X. _V ) ) = ( ( A i^i ( B X. _V ) ) i^i ( C X. _V ) )
4		xpindir 4866	. . . 4 |- ( ( B i^i C ) X. _V ) = ( ( B X. _V ) i^i ( C X. _V ) )
5	4	ineq2i 3405	. . 3 |- ( A i^i ( ( B i^i C ) X. _V ) ) = ( A i^i ( ( B X. _V ) i^i ( C X. _V ) ) )
6		df-res 4737	. . 3 |- ( A |` ( B i^i C ) ) = ( A i^i ( ( B i^i C ) X. _V ) )
7		inass 3417	. . 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 2263	. 2 |- ( ( A i^i ( B X. _V ) ) i^i ( C X. _V ) ) = ( A |` ( B i^i C ) )
9	1, 3, 8	3eqtri 2256	1 |- ( ( A |` B ) |` C ) = ( A |` ( B i^i C ) )

Syntax hints:   = wceq 1397   _Vcvv 2802   i^i cin 3199   X. cxp 4723   |` cres 4727

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-rex 2516  df-v 2804  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-opab 4151  df-xp 4731  df-rel 4732  df-res 4737

This theorem is referenced by:  rescom  5038  resabs1  5042  resima2  5047  resmpt3  5062  resdisj  5165  rescnvcnv  5199  funimaexg  5414  fresin  5515  resdif  5605  pmresg  6845  setsslid  13134