Theorem resiun1OLD 5417

Description: Obsolete proof of resiun1 5416 as of 25-Aug-2021. (Contributed by Mario Carneiro, 29-May-2015.) (New usage is discouraged.) (Proof modification is discouraged.)

Assertion

Ref	Expression
resiun1OLD	|- ( 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 resiun1OLD

Step	Hyp	Ref	Expression
1		iunin2 4584	. 2 |- U_ x e. A ( ( C X. _V ) i^i B ) = ( ( C X. _V ) i^i U_ x e. A B )
2		df-res 5126	. . . . 5 |- ( B |` C ) = ( B i^i ( C X. _V ) )
3		incom 3805	. . . . 5 |- ( B i^i ( C X. _V ) ) = ( ( C X. _V ) i^i B )
4	2, 3	eqtri 2644	. . . 4 |- ( B |` C ) = ( ( C X. _V ) i^i B )
5	4	a1i 11	. . 3 |- ( x e. A -> ( B |` C ) = ( ( C X. _V ) i^i B ) )
6	5	iuneq2i 4539	. 2 |- U_ x e. A ( B |` C ) = U_ x e. A ( ( C X. _V ) i^i B )
7		df-res 5126	. . 3 |- ( U_ x e. A B |` C ) = ( U_ x e. A B i^i ( C X. _V ) )
8		incom 3805	. . 3 |- ( U_ x e. A B i^i ( C X. _V ) ) = ( ( C X. _V ) i^i U_ x e. A B )
9	7, 8	eqtri 2644	. 2 |- ( U_ x e. A B |` C ) = ( ( C X. _V ) i^i U_ x e. A B )
10	1, 6, 9	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)