Theorem resundir 4942

Description: Distributive law for restriction over union. (Contributed by NM, 23-Sep-2004.)

Assertion

Ref	Expression
resundir	|- ( ( A u. B ) |` C ) = ( ( A |` C ) u. ( B |` C ) )

Proof of Theorem resundir

Step	Hyp	Ref	Expression
1		indir 3399	. 2 |- ( ( A u. B ) i^i ( C X. _V ) ) = ( ( A i^i ( C X. _V ) ) u. ( B i^i ( C X. _V ) ) )
2		df-res 4659	. 2 |- ( ( A u. B ) |` C ) = ( ( A u. B ) i^i ( C X. _V ) )
3		df-res 4659	. . 3 |- ( A |` C ) = ( A i^i ( C X. _V ) )
4		df-res 4659	. . 3 |- ( B |` C ) = ( B i^i ( C X. _V ) )
5	3, 4	uneq12i 3302	. 2 |- ( ( A |` C ) u. ( B |` C ) ) = ( ( A i^i ( C X. _V ) ) u. ( B i^i ( C X. _V ) ) )
6	1, 2, 5	3eqtr4i 2220	1 |- ( ( A u. B ) |` C ) = ( ( A |` C ) u. ( B |` C ) )

Syntax hints:   = wceq 1364   _Vcvv 2752   u. cun 3142   i^i cin 3143   X. cxp 4645   |` cres 4649

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-v 2754  df-un 3148  df-in 3150  df-res 4659

This theorem is referenced by:  imaundir  5063  fvunsng  5734  fvsnun1  5737  fvsnun2  5738  fsnunfv  5741  fsnunres  5742  fseq1p1m1  10131  setsresg  12562  setscom  12564  setsslid  12575