Theorem resiun1 4919

Description: Distribution of restriction over indexed union. (Contributed by Mario Carneiro, 29-May-2015.)

Assertion

Ref	Expression
resiun1	|- ( U_ x e. A B |` C ) = U_ x e. A ( B |` C )

Distinct variable group:   x, C

Allowed substitution hints:   A( x)   B( x)

Proof of Theorem resiun1

Step	Hyp	Ref	Expression
1		iunin2 3945	. 2 |- U_ x e. A ( ( C X. _V ) i^i B ) = ( ( C X. _V ) i^i U_ x e. A B )
2		df-res 4632	. . . . 5 |- ( B |` C ) = ( B i^i ( C X. _V ) )
3		incom 3325	. . . . 5 |- ( B i^i ( C X. _V ) ) = ( ( C X. _V ) i^i B )
4	2, 3	eqtri 2196	. . . 4 |- ( B |` C ) = ( ( C X. _V ) i^i B )
5	4	a1i 9	. . 3 |- ( x e. A -> ( B |` C ) = ( ( C X. _V ) i^i B ) )
6	5	iuneq2i 3900	. 2 |- U_ x e. A ( B |` C ) = U_ x e. A ( ( C X. _V ) i^i B )
7		df-res 4632	. . 3 |- ( U_ x e. A B |` C ) = ( U_ x e. A B i^i ( C X. _V ) )
8		incom 3325	. . 3 |- ( U_ x e. A B i^i ( C X. _V ) ) = ( ( C X. _V ) i^i U_ x e. A B )
9	7, 8	eqtri 2196	. 2 |- ( U_ x e. A B |` C ) = ( ( C X. _V ) i^i U_ x e. A B )
10	1, 6, 9	3eqtr4ri 2207	1 |- ( U_ x e. A B |` C ) = U_ x e. A ( B |` C )

Syntax hints:   = wceq 1353   e. wcel 2146   _Vcvv 2735   i^i cin 3126   U_ciun 3882   X. cxp 4618   |` cres 4622

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 709  ax-5 1445  ax-7 1446  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-10 1503  ax-11 1504  ax-i12 1505  ax-bndl 1507  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532  ax-i5r 1533  ax-ext 2157

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1459  df-sb 1761  df-clab 2162  df-cleq 2168  df-clel 2171  df-nfc 2306  df-ral 2458  df-rex 2459  df-v 2737  df-in 3133  df-ss 3140  df-iun 3884  df-res 4632

This theorem is referenced by: (None)