Theorem resiun1 5416

Description: Distribution of restriction over indexed union. (Contributed by Mario Carneiro, 29-May-2015.) (Proof shortened by JJ, 25-Aug-2021.)

Assertion

Ref	Expression
resiun1	|- ( U_ x e. A B |` C ) = U_ x e. A ( B |` C )

Distinct variable group:   x, C

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

Proof of Theorem resiun1

Step	Hyp	Ref	Expression
1		iunin1 4585	. 2 |- U_ x e. A ( B i^i ( C X. _V ) ) = ( U_ x e. A B i^i ( C X. _V ) )
2		df-res 5126	. . . 4 |- ( B |` C ) = ( B i^i ( C X. _V ) )
3	2	a1i 11	. . 3 |- ( x e. A -> ( B |` C ) = ( B i^i ( C X. _V ) ) )
4	3	iuneq2i 4539	. 2 |- U_ x e. A ( B |` C ) = U_ x e. A ( B i^i ( C X. _V ) )
5		df-res 5126	. 2 |- ( U_ x e. A B |` C ) = ( U_ x e. A B i^i ( C X. _V ) )
6	1, 4, 5	3eqtr4ri 2655	1 |- ( U_ x e. A B |` C ) = U_ x e. A ( B |` C )

Syntax hints:   = wceq 1483   e. wcel 1990   _Vcvv 3200   i^i cin 3573   U_ciun 4520   X. cxp 5112   |` cres 5116

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  df-res 5126

This theorem is referenced by: (None)