Theorem iinuni 4609

Description: A relationship involving union and indexed intersection. Exercise 23 of [Enderton] p. 33. (Contributed by NM, 25-Nov-2003.) (Proof shortened by Mario Carneiro, 17-Nov-2016.)

Assertion

Ref	Expression
iinuni	|- ( A u. |^| B ) = |^|_ x e. B ( A u. x )

Distinct variable groups:   x, A   x, B

Proof of Theorem iinuni

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		r19.32v 3083	. . . 4 |- ( A. x e. B ( y e. A \/ y e. x ) <-> ( y e. A \/ A. x e. B y e. x ) )
2		elun 3753	. . . . 5 |- ( y e. ( A u. x ) <-> ( y e. A \/ y e. x ) )
3	2	ralbii 2980	. . . 4 |- ( A. x e. B y e. ( A u. x ) <-> A. x e. B ( y e. A \/ y e. x ) )
4		vex 3203	. . . . . 6 |- y e. _V
5	4	elint2 4482	. . . . 5 |- ( y e. |^| B <-> A. x e. B y e. x )
6	5	orbi2i 541	. . . 4 |- ( ( y e. A \/ y e. |^| B ) <-> ( y e. A \/ A. x e. B y e. x ) )
7	1, 3, 6	3bitr4ri 293	. . 3 |- ( ( y e. A \/ y e. |^| B ) <-> A. x e. B y e. ( A u. x ) )
8	7	abbii 2739	. 2 |- { y | ( y e. A \/ y e. |^| B ) } = { y | A. x e. B y e. ( A u. x ) }
9		df-un 3579	. 2 |- ( A u. |^| B ) = { y | ( y e. A \/ y e. |^| B ) }
10		df-iin 4523	. 2 |- |^|_ x e. B ( A u. x ) = { y | A. x e. B y e. ( A u. x ) }
11	8, 9, 10	3eqtr4i 2654	1 |- ( A u. |^| B ) = |^|_ x e. B ( A u. x )

Syntax hints:   \/ wo 383   = wceq 1483   e. wcel 1990   {cab 2608   A.wral 2912   u. cun 3572   |^|cint 4475   |^|_ciin 4521

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-v 3202  df-un 3579  df-int 4476  df-iin 4523

This theorem is referenced by: (None)