Theorem rescom 5423

Description: Commutative law for restriction. (Contributed by NM, 27-Mar-1998.)

Assertion

Ref	Expression
rescom	|- ( ( A |` B ) |` C ) = ( ( A |` C ) |` B )

Proof of Theorem rescom

Step	Hyp	Ref	Expression
1		incom 3805	. . 3 |- ( B i^i C ) = ( C i^i B )
2	1	reseq2i 5393	. 2 |- ( A |` ( B i^i C ) ) = ( A |` ( C i^i B ) )
3		resres 5409	. 2 |- ( ( A |` B ) |` C ) = ( A |` ( B i^i C ) )
4		resres 5409	. 2 |- ( ( A |` C ) |` B ) = ( A |` ( C i^i B ) )
5	2, 3, 4	3eqtr4i 2654	1 |- ( ( A |` B ) |` C ) = ( ( A |` C ) |` B )

Syntax hints:   = wceq 1483   i^i cin 3573   |` cres 5116

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-opab 4713  df-xp 5120  df-rel 5121  df-res 5126

This theorem is referenced by:  resabs2  5429  setscom  15903  dvres3a  23678  cpnres  23700  dvmptres3  23719  limsupresuz  39935  liminfresuz  40016