Theorem iunab 4013

Description: The indexed union of a class abstraction. (Contributed by NM, 27-Dec-2004.)

Assertion

Ref	Expression
iunab	|- U_x e. A {y | ph} = {y | E.x e. A ph}

Distinct variable groups:   y,A   x,y

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

Proof of Theorem iunab

Step	Hyp	Ref	Expression
1		nfcv 2490	. . . 4 |- F/_yA
2		nfab1 2492	. . . 4 |- F/_y{y | ph}
3	1, 2	nfiun 3996	. . 3 |- F/_yU_x e. A {y | ph}
4		nfab1 2492	. . 3 |- F/_y{y | E.x e. A ph}
5	3, 4	cleqf 2514	. 2 |- (U_x e. A {y | ph} = {y | E.x e. A ph} <-> A.y(y e. U_x e. A {y | ph} <-> y e. {y | E.x e. A ph}))
6		abid 2341	. . . 4 |- (y e. {y | ph} <-> ph)
7	6	rexbii 2640	. . 3 |- (E.x e. A y e. {y | ph} <-> E.x e. A ph)
8		eliun 3974	. . 3 |- (y e. U_x e. A {y | ph} <-> E.x e. A y e. {y | ph})
9		abid 2341	. . 3 |- (y e. {y | E.x e. A ph} <-> E.x e. A ph)
10	7, 8, 9	3bitr4i 268	. 2 |- (y e. U_x e. A {y | ph} <-> y e. {y | E.x e. A ph})
11	5, 10	mpgbir 1550	1 |- U_x e. A {y | ph} = {y | E.x e. A ph}

Syntax hints:   <-> wb 176   = wceq 1642   e. wcel 1710  {cab 2339  E.wrex 2616  U_ciun 3970

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-ral 2620  df-rex 2621  df-v 2862  df-iun 3972

This theorem is referenced by:  iunrab  4014  iunid  4022  dfimafn2  5368