Theorem resiun1 4961

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 3976	. 2 |- U_ x e. A ( ( C X. _V ) i^i B ) = ( ( C X. _V ) i^i U_ x e. A B )
2		df-res 4671	. . . . 5 |- ( B |` C ) = ( B i^i ( C X. _V ) )
3		incom 3351	. . . . 5 |- ( B i^i ( C X. _V ) ) = ( ( C X. _V ) i^i B )
4	2, 3	eqtri 2214	. . . 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 3930	. 2 |- U_ x e. A ( B |` C ) = U_ x e. A ( ( C X. _V ) i^i B )
7		df-res 4671	. . 3 |- ( U_ x e. A B |` C ) = ( U_ x e. A B i^i ( C X. _V ) )
8		incom 3351	. . 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 2214	. 2 |- ( U_ x e. A B |` C ) = ( ( C X. _V ) i^i U_ x e. A B )
10	1, 6, 9	3eqtr4ri 2225	1 |- ( U_ x e. A B |` C ) = U_ x e. A ( B |` C )

Syntax hints:   = wceq 1364   e. wcel 2164   _Vcvv 2760   i^i cin 3152   U_ciun 3912   X. cxp 4657   |` cres 4661

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 2175

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-v 2762  df-in 3159  df-ss 3166  df-iun 3914  df-res 4671

This theorem is referenced by: (None)