Theorem resiun2 4911

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

Assertion

Ref	Expression
resiun2	|- ( C |` U_ x e. A B ) = U_ x e. A ( C |` B )

Distinct variable group:   x, C

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

Proof of Theorem resiun2

Step	Hyp	Ref	Expression
1		df-res 4623	. 2 |- ( C |` U_ x e. A B ) = ( C i^i ( U_ x e. A B X. _V ) )
2		df-res 4623	. . . . 5 |- ( C |` B ) = ( C i^i ( B X. _V ) )
3	2	a1i 9	. . . 4 |- ( x e. A -> ( C |` B ) = ( C i^i ( B X. _V ) ) )
4	3	iuneq2i 3891	. . 3 |- U_ x e. A ( C |` B ) = U_ x e. A ( C i^i ( B X. _V ) )
5		xpiundir 4670	. . . . 5 |- ( U_ x e. A B X. _V ) = U_ x e. A ( B X. _V )
6	5	ineq2i 3325	. . . 4 |- ( C i^i ( U_ x e. A B X. _V ) ) = ( C i^i U_ x e. A ( B X. _V ) )
7		iunin2 3936	. . . 4 |- U_ x e. A ( C i^i ( B X. _V ) ) = ( C i^i U_ x e. A ( B X. _V ) )
8	6, 7	eqtr4i 2194	. . 3 |- ( C i^i ( U_ x e. A B X. _V ) ) = U_ x e. A ( C i^i ( B X. _V ) )
9	4, 8	eqtr4i 2194	. 2 |- U_ x e. A ( C |` B ) = ( C i^i ( U_ x e. A B X. _V ) )
10	1, 9	eqtr4i 2194	1 |- ( C |` U_ x e. A B ) = U_ x e. A ( C |` B )

Syntax hints:   = wceq 1348   e. wcel 2141   _Vcvv 2730   i^i cin 3120   U_ciun 3873   X. cxp 4609   |` cres 4613

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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-iun 3875  df-opab 4051  df-xp 4617  df-res 4623

This theorem is referenced by: (None)