Theorem resiun1 4699

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 3776	. 2 |- U_ x e. A ( ( C X. _V ) i^i B ) = ( ( C X. _V ) i^i U_ x e. A B )
2		df-res 4423	. . . . 5 |- ( B |` C ) = ( B i^i ( C X. _V ) )
3		incom 3181	. . . . 5 |- ( B i^i ( C X. _V ) ) = ( ( C X. _V ) i^i B )
4	2, 3	eqtri 2105	. . . 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 3731	. 2 |- U_ x e. A ( B |` C ) = U_ x e. A ( ( C X. _V ) i^i B )
7		df-res 4423	. . 3 |- ( U_ x e. A B |` C ) = ( U_ x e. A B i^i ( C X. _V ) )
8		incom 3181	. . 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 2105	. 2 |- ( U_ x e. A B |` C ) = ( ( C X. _V ) i^i U_ x e. A B )
10	1, 6, 9	3eqtr4ri 2116	1 |- ( U_ x e. A B |` C ) = U_ x e. A ( B |` C )

Syntax hints:   = wceq 1287   e. wcel 1436   _Vcvv 2615   i^i cin 2987   U_ciun 3713   X. cxp 4409   |` cres 4413

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067

This theorem depends on definitions:  df-bi 115  df-tru 1290  df-nf 1393  df-sb 1690  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-ral 2360  df-rex 2361  df-v 2617  df-in 2994  df-ss 3001  df-iun 3715  df-res 4423

This theorem is referenced by: (None)