Theorem resiun2 4631

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 4357	. 2 |- ( C |` U_ x e. A B ) = ( C i^i ( U_ x e. A B X. _V ) )
2		df-res 4357	. . . . 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 3675	. . 3 |- U_ x e. A ( C |` B ) = U_ x e. A ( C i^i ( B X. _V ) )
5		xpiundir 4399	. . . . 5 |- ( U_ x e. A B X. _V ) = U_ x e. A ( B X. _V )
6	5	ineq2i 3135	. . . 4 |- ( C i^i ( U_ x e. A B X. _V ) ) = ( C i^i U_ x e. A ( B X. _V ) )
7		iunin2 3720	. . . 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 2063	. . 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 2063	. 2 |- U_ x e. A ( C |` B ) = ( C i^i ( U_ x e. A B X. _V ) )
10	1, 9	eqtr4i 2063	1 |- ( C |` U_ x e. A B ) = U_ x e. A ( C |` B )

Syntax hints:   = wceq 1243   e. wcel 1393   _Vcvv 2557   i^i cin 2916   U_ciun 3657   X. cxp 4343   |` cres 4347

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944

This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-iun 3659  df-opab 3819  df-xp 4351  df-res 4357

This theorem is referenced by: (None)