Theorem resundi 5076

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 4832	. . . 4 |- ( ( B u. C ) X. _V ) = ( ( B X. _V ) u. ( C X. _V ) )
2	1	ineq2i 3429	. . 3 |- ( A i^i ( ( B u. C ) X. _V ) ) = ( A i^i ( ( B X. _V ) u. ( C X. _V ) ) )
3		indi 3478	. . 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 2259	. 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 4786	. 2 |- ( A |` ( B u. C ) ) = ( A i^i ( ( B u. C ) X. _V ) )
6		df-res 4786	. . 3 |- ( A |` B ) = ( A i^i ( B X. _V ) )
7		df-res 4786	. . 3 |- ( A |` C ) = ( A i^i ( C X. _V ) )
8	6, 7	uneq12i 3381	. 2 |- ( ( A |` B ) u. ( A |` C ) ) = ( ( A i^i ( B X. _V ) ) u. ( A i^i ( C X. _V ) ) )
9	4, 5, 8	3eqtr4i 2269	1 |- ( A |` ( B u. C ) ) = ( ( A |` B ) u. ( A |` C ) )

This proof depends on syntax axioms:   = wceq 1402   _Vcvv 2821   u. cun 3218   i^i cin 3219   X. cxp 4772   |` cres 4776

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This proof depends on definitions:  df-bi 117  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-v 2823  df-un 3224  df-in 3226  df-opab 4193  df-xp 4780  df-res 4786

This theorem is used by:  imaundi  5200  relresfld  5317  relcoi1  5319  resasplitss  5569  fnsnsplitss  5914  fnsnsplitdc  6778  mapunen  7151  fnfi  7250  fseq1p1m1  10501  resunimafz0  11274