Theorem resres 5041

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 4752	. 2 |- ( ( A |` B ) |` C ) = ( ( A |` B ) i^i ( C X. _V ) )
2		df-res 4752	. . 3 |- ( A |` B ) = ( A i^i ( B X. _V ) )
3	2	ineq1i 3415	. 2 |- ( ( A |` B ) i^i ( C X. _V ) ) = ( ( A i^i ( B X. _V ) ) i^i ( C X. _V ) )
4		xpindir 4882	. . . 4 |- ( ( B i^i C ) X. _V ) = ( ( B X. _V ) i^i ( C X. _V ) )
5	4	ineq2i 3416	. . 3 |- ( A i^i ( ( B i^i C ) X. _V ) ) = ( A i^i ( ( B X. _V ) i^i ( C X. _V ) ) )
6		df-res 4752	. . 3 |- ( A |` ( B i^i C ) ) = ( A i^i ( ( B i^i C ) X. _V ) )
7		inass 3428	. . 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 2264	. 2 |- ( ( A i^i ( B X. _V ) ) i^i ( C X. _V ) ) = ( A |` ( B i^i C ) )
9	1, 3, 8	3eqtri 2257	1 |- ( ( A |` B ) |` C ) = ( A |` ( B i^i C ) )

Syntax hints:   = wceq 1398   _Vcvv 2812   i^i cin 3209   X. cxp 4738   |` cres 4742

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2206  ax-ext 2214  ax-sep 4221  ax-pow 4279  ax-pr 4314

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ral 2525  df-rex 2526  df-v 2814  df-un 3214  df-in 3216  df-ss 3223  df-pw 3667  df-sn 3688  df-pr 3689  df-op 3691  df-opab 4165  df-xp 4746  df-rel 4747  df-res 4752

This theorem is referenced by:  rescom  5054  resabs1  5058  resima2  5063  resmpt3  5078  resdisj  5182  rescnvcnv  5216  funimaexg  5431  fresin  5534  resdif  5627  pmresg  6901  setsslid  13237