Theorem iunin1 4585

Description: Indexed union of intersection. Generalization of half of theorem "Distributive laws" in [Enderton] p. 30. Use uniiun 4573 to recover Enderton's theorem. (Contributed by Mario Carneiro, 30-Aug-2015.)

Assertion

Ref	Expression
iunin1	|- U_ x e. A ( C i^i B ) = ( U_ x e. A C i^i B )

Distinct variable group:   x, B

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

Proof of Theorem iunin1

Step	Hyp	Ref	Expression
1		iunin2 4584	. 2 |- U_ x e. A ( B i^i C ) = ( B i^i U_ x e. A C )
2		incom 3805	. . . 4 |- ( C i^i B ) = ( B i^i C )
3	2	a1i 11	. . 3 |- ( x e. A -> ( C i^i B ) = ( B i^i C ) )
4	3	iuneq2i 4539	. 2 |- U_ x e. A ( C i^i B ) = U_ x e. A ( B i^i C )
5		incom 3805	. 2 |- ( U_ x e. A C i^i B ) = ( B i^i U_ x e. A C )
6	1, 4, 5	3eqtr4i 2654	1 |- U_ x e. A ( C i^i B ) = ( U_ x e. A C i^i B )

Syntax hints:   = wceq 1483   e. wcel 1990   i^i cin 3573   U_ciun 4520

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-v 3202  df-in 3581  df-ss 3588  df-iun 4522

This theorem is referenced by:  2iunin  4588  resiun1  5416  tgrest  20963  metnrmlem3  22664  limciun  23658  uniin1  29367  disjunsn  29407  measinblem  30283  sstotbnd2  33573  subsaliuncl  40576  sge0iunmptlemre  40632