Theorem resiun2 5025

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 4731	. 2 |- ( C |` U_ x e. A B ) = ( C i^i ( U_ x e. A B X. _V ) )
2		df-res 4731	. . . . 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 3983	. . 3 |- U_ x e. A ( C |` B ) = U_ x e. A ( C i^i ( B X. _V ) )
5		xpiundir 4778	. . . . 5 |- ( U_ x e. A B X. _V ) = U_ x e. A ( B X. _V )
6	5	ineq2i 3402	. . . 4 |- ( C i^i ( U_ x e. A B X. _V ) ) = ( C i^i U_ x e. A ( B X. _V ) )
7		iunin2 4029	. . . 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 2253	. . 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 2253	. 2 |- U_ x e. A ( C |` B ) = ( C i^i ( U_ x e. A B X. _V ) )
10	1, 9	eqtr4i 2253	1 |- ( C |` U_ x e. A B ) = U_ x e. A ( C |` B )

Syntax hints:   = wceq 1395   e. wcel 2200   _Vcvv 2799   i^i cin 3196   U_ciun 3965   X. cxp 4717   |` cres 4721

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-14 2203  ax-ext 2211  ax-sep 4202  ax-pow 4258  ax-pr 4293

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-v 2801  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-iun 3967  df-opab 4146  df-xp 4725  df-res 4731

This theorem is referenced by: (None)