Theorem iunss 4008

Description: Subset theorem for an indexed union. (Contributed by NM, 13-Sep-2003.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)

Assertion

Ref	Expression
iunss	|- (U_x e. A B C_ C <-> A.x e. A B C_ C)

Distinct variable group:   x,C

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

Proof of Theorem iunss

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-iun 3972	. . 3 |- U_x e. A B = {y | E.x e. A y e. B}
2	1	sseq1i 3296	. 2 |- (U_x e. A B C_ C <-> {y | E.x e. A y e. B} C_ C)
3		abss 3336	. 2 |- ({y | E.x e. A y e. B} C_ C <-> A.y(E.x e. A y e. B -> y e. C))
4		dfss2 3263	. . . 4 |- (B C_ C <-> A.y(y e. B -> y e. C))
5	4	ralbii 2639	. . 3 |- (A.x e. A B C_ C <-> A.x e. A A.y(y e. B -> y e. C))
6		ralcom4 2878	. . 3 |- (A.x e. A A.y(y e. B -> y e. C) <-> A.yA.x e. A (y e. B -> y e. C))
7		r19.23v 2731	. . . 4 |- (A.x e. A (y e. B -> y e. C) <-> (E.x e. A y e. B -> y e. C))
8	7	albii 1566	. . 3 |- (A.yA.x e. A (y e. B -> y e. C) <-> A.y(E.x e. A y e. B -> y e. C))
9	5, 6, 8	3bitrri 263	. 2 |- (A.y(E.x e. A y e. B -> y e. C) <-> A.x e. A B C_ C)
10	2, 3, 9	3bitri 262	1 |- (U_x e. A B C_ C <-> A.x e. A B C_ C)

Syntax hints:   -> wi 4   <-> wb 176  A.wal 1540   e. wcel 1710  {cab 2339  A.wral 2615  E.wrex 2616   C_ wss 3258  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-nan 1288  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-nin 3212  df-compl 3213  df-in 3214  df-ss 3260  df-iun 3972

This theorem is referenced by:  iunss2  4012