Theorem rabiun 33382

Description: Abstraction restricted to an indexed union. (Contributed by Brendan Leahy, 26-Oct-2017.)

Assertion

Ref	Expression
rabiun	|- { x e. U_ y e. A B | ph } = U_ y e. A { x e. B | ph }

Distinct variable groups:   ph, y   x, A   x, y

Allowed substitution hints:   ph( x)   A( y)   B( x, y)

Proof of Theorem rabiun

Step	Hyp	Ref	Expression
1		eliun 4524	. . . . . 6 |- ( x e. U_ y e. A B <-> E. y e. A x e. B )
2	1	anbi1i 731	. . . . 5 |- ( ( x e. U_ y e. A B /\ ph ) <-> ( E. y e. A x e. B /\ ph ) )
3		r19.41v 3089	. . . . 5 |- ( E. y e. A ( x e. B /\ ph ) <-> ( E. y e. A x e. B /\ ph ) )
4	2, 3	bitr4i 267	. . . 4 |- ( ( x e. U_ y e. A B /\ ph ) <-> E. y e. A ( x e. B /\ ph ) )
5	4	abbii 2739	. . 3 |- { x | ( x e. U_ y e. A B /\ ph ) } = { x | E. y e. A ( x e. B /\ ph ) }
6		df-rab 2921	. . 3 |- { x e. U_ y e. A B | ph } = { x | ( x e. U_ y e. A B /\ ph ) }
7		iunab 4566	. . 3 |- U_ y e. A { x | ( x e. B /\ ph ) } = { x | E. y e. A ( x e. B /\ ph ) }
8	5, 6, 7	3eqtr4i 2654	. 2 |- { x e. U_ y e. A B | ph } = U_ y e. A { x | ( x e. B /\ ph ) }
9		df-rab 2921	. . . 4 |- { x e. B | ph } = { x | ( x e. B /\ ph ) }
10	9	a1i 11	. . 3 |- ( y e. A -> { x e. B | ph } = { x | ( x e. B /\ ph ) } )
11	10	iuneq2i 4539	. 2 |- U_ y e. A { x e. B | ph } = U_ y e. A { x | ( x e. B /\ ph ) }
12	8, 11	eqtr4i 2647	1 |- { x e. U_ y e. A B | ph } = U_ y e. A { x e. B | ph }

Syntax hints:   /\ wa 384   = wceq 1483   e. wcel 1990   {cab 2608   E.wrex 2913   {crab 2916   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-rab 2921  df-v 3202  df-in 3581  df-ss 3588  df-iun 4522

This theorem is referenced by:  itg2addnclem2  33462