Theorem resiun2 4963

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 4672	. 2 |- ( C |` U_ x e. A B ) = ( C i^i ( U_ x e. A B X. _V ) )
2		df-res 4672	. . . . 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 3931	. . 3 |- U_ x e. A ( C |` B ) = U_ x e. A ( C i^i ( B X. _V ) )
5		xpiundir 4719	. . . . 5 |- ( U_ x e. A B X. _V ) = U_ x e. A ( B X. _V )
6	5	ineq2i 3358	. . . 4 |- ( C i^i ( U_ x e. A B X. _V ) ) = ( C i^i U_ x e. A ( B X. _V ) )
7		iunin2 3977	. . . 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 2217	. . 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 2217	. 2 |- U_ x e. A ( C |` B ) = ( C i^i ( U_ x e. A B X. _V ) )
10	1, 9	eqtr4i 2217	1 |- ( C |` U_ x e. A B ) = U_ x e. A ( C |` B )

Syntax hints:   = wceq 1364   e. wcel 2164   _Vcvv 2760   i^i cin 3153   U_ciun 3913   X. cxp 4658   |` cres 4662

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2167  ax-ext 2175  ax-sep 4148  ax-pow 4204  ax-pr 4239

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-v 2762  df-un 3158  df-in 3160  df-ss 3167  df-pw 3604  df-sn 3625  df-pr 3626  df-op 3628  df-iun 3915  df-opab 4092  df-xp 4666  df-res 4672

This theorem is referenced by: (None)