Theorem resres 4903

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 4623	. 2 |- ( ( A |` B ) |` C ) = ( ( A |` B ) i^i ( C X. _V ) )
2		df-res 4623	. . 3 |- ( A |` B ) = ( A i^i ( B X. _V ) )
3	2	ineq1i 3324	. 2 |- ( ( A |` B ) i^i ( C X. _V ) ) = ( ( A i^i ( B X. _V ) ) i^i ( C X. _V ) )
4		xpindir 4747	. . . 4 |- ( ( B i^i C ) X. _V ) = ( ( B X. _V ) i^i ( C X. _V ) )
5	4	ineq2i 3325	. . 3 |- ( A i^i ( ( B i^i C ) X. _V ) ) = ( A i^i ( ( B X. _V ) i^i ( C X. _V ) ) )
6		df-res 4623	. . 3 |- ( A |` ( B i^i C ) ) = ( A i^i ( ( B i^i C ) X. _V ) )
7		inass 3337	. . 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 2202	. 2 |- ( ( A i^i ( B X. _V ) ) i^i ( C X. _V ) ) = ( A |` ( B i^i C ) )
9	1, 3, 8	3eqtri 2195	1 |- ( ( A |` B ) |` C ) = ( A |` ( B i^i C ) )

Syntax hints:   = wceq 1348   _Vcvv 2730   i^i cin 3120   X. cxp 4609   |` cres 4613

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-opab 4051  df-xp 4617  df-rel 4618  df-res 4623

This theorem is referenced by:  rescom  4916  resabs1  4920  resima2  4925  resmpt3  4940  resdisj  5039  rescnvcnv  5073  funimaexg  5282  fresin  5376  resdif  5464  pmresg  6654  setsslid  12466