Theorem resiun1 4903

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 3929	. 2 |- U_ x e. A ( ( C X. _V ) i^i B ) = ( ( C X. _V ) i^i U_ x e. A B )
2		df-res 4616	. . . . 5 |- ( B |` C ) = ( B i^i ( C X. _V ) )
3		incom 3314	. . . . 5 |- ( B i^i ( C X. _V ) ) = ( ( C X. _V ) i^i B )
4	2, 3	eqtri 2186	. . . 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 3884	. 2 |- U_ x e. A ( B |` C ) = U_ x e. A ( ( C X. _V ) i^i B )
7		df-res 4616	. . 3 |- ( U_ x e. A B |` C ) = ( U_ x e. A B i^i ( C X. _V ) )
8		incom 3314	. . 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 2186	. 2 |- ( U_ x e. A B |` C ) = ( ( C X. _V ) i^i U_ x e. A B )
10	1, 6, 9	3eqtr4ri 2197	1 |- ( U_ x e. A B |` C ) = U_ x e. A ( B |` C )

Syntax hints:   = wceq 1343   e. wcel 2136   _Vcvv 2726   i^i cin 3115   U_ciun 3866   X. cxp 4602   |` cres 4606

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 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-rex 2450  df-v 2728  df-in 3122  df-ss 3129  df-iun 3868  df-res 4616

This theorem is referenced by: (None)