Theorem resundi 4891

Description: Distributive law for restriction over union. Theorem 31 of [Suppes] p. 65. (Contributed by NM, 30-Sep-2002.)

Assertion

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

Proof of Theorem resundi

Step	Hyp	Ref	Expression
1		xpundir 4655	. . . 4 |- ( ( B u. C ) X. _V ) = ( ( B X. _V ) u. ( C X. _V ) )
2	1	ineq2i 3315	. . 3 |- ( A i^i ( ( B u. C ) X. _V ) ) = ( A i^i ( ( B X. _V ) u. ( C X. _V ) ) )
3		indi 3364	. . 3 |- ( A i^i ( ( B X. _V ) u. ( C X. _V ) ) ) = ( ( A i^i ( B X. _V ) ) u. ( A i^i ( C X. _V ) ) )
4	2, 3	eqtri 2185	. 2 |- ( A i^i ( ( B u. C ) X. _V ) ) = ( ( A i^i ( B X. _V ) ) u. ( A i^i ( C X. _V ) ) )
5		df-res 4610	. 2 |- ( A |` ( B u. C ) ) = ( A i^i ( ( B u. C ) X. _V ) )
6		df-res 4610	. . 3 |- ( A |` B ) = ( A i^i ( B X. _V ) )
7		df-res 4610	. . 3 |- ( A |` C ) = ( A i^i ( C X. _V ) )
8	6, 7	uneq12i 3269	. 2 |- ( ( A |` B ) u. ( A |` C ) ) = ( ( A i^i ( B X. _V ) ) u. ( A i^i ( C X. _V ) ) )
9	4, 5, 8	3eqtr4i 2195	1 |- ( A |` ( B u. C ) ) = ( ( A |` B ) u. ( A |` C ) )

Syntax hints:   = wceq 1342   _Vcvv 2721   u. cun 3109   i^i cin 3110   X. cxp 4596   |` cres 4600

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 699  ax-5 1434  ax-7 1435  ax-gen 1436  ax-ie1 1480  ax-ie2 1481  ax-8 1491  ax-10 1492  ax-11 1493  ax-i12 1494  ax-bndl 1496  ax-4 1497  ax-17 1513  ax-i9 1517  ax-ial 1521  ax-i5r 1522  ax-ext 2146

This theorem depends on definitions:  df-bi 116  df-tru 1345  df-nf 1448  df-sb 1750  df-clab 2151  df-cleq 2157  df-clel 2160  df-nfc 2295  df-v 2723  df-un 3115  df-in 3117  df-opab 4038  df-xp 4604  df-res 4610

This theorem is referenced by:  imaundi  5010  relresfld  5127  relcoi1  5129  resasplitss  5361  fnsnsplitss  5678  fnsnsplitdc  6464  fnfi  6893  fseq1p1m1  10019  resunimafz0  10730