Theorem resundi 4991

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 4750	. . . 4 |- ( ( B u. C ) X. _V ) = ( ( B X. _V ) u. ( C X. _V ) )
2	1	ineq2i 3379	. . 3 |- ( A i^i ( ( B u. C ) X. _V ) ) = ( A i^i ( ( B X. _V ) u. ( C X. _V ) ) )
3		indi 3428	. . 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 2228	. 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 4705	. 2 |- ( A |` ( B u. C ) ) = ( A i^i ( ( B u. C ) X. _V ) )
6		df-res 4705	. . 3 |- ( A |` B ) = ( A i^i ( B X. _V ) )
7		df-res 4705	. . 3 |- ( A |` C ) = ( A i^i ( C X. _V ) )
8	6, 7	uneq12i 3333	. 2 |- ( ( A |` B ) u. ( A |` C ) ) = ( ( A i^i ( B X. _V ) ) u. ( A i^i ( C X. _V ) ) )
9	4, 5, 8	3eqtr4i 2238	1 |- ( A |` ( B u. C ) ) = ( ( A |` B ) u. ( A |` C ) )

Syntax hints:   = wceq 1373   _Vcvv 2776   u. cun 3172   i^i cin 3173   X. cxp 4691   |` cres 4695

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2189

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-v 2778  df-un 3178  df-in 3180  df-opab 4122  df-xp 4699  df-res 4705

This theorem is referenced by:  imaundi  5114  relresfld  5231  relcoi1  5233  resasplitss  5477  fnsnsplitss  5806  fnsnsplitdc  6614  fnfi  7064  fseq1p1m1  10251  resunimafz0  11013