Theorem iunxdif2 4363

Description: Indexed union with a class difference as its index. (Contributed by NM, 10-Dec-2004.)

Hypothesis

Ref	Expression
iunxdif2.1	|- ( x = y -> C = D )

Assertion

Ref	Expression
iunxdif2	|- ( A. x e. A E. y e. ( A \ B ) C C_ D -> U_ y e. ( A \ B ) D = U_ x e. A C )

Distinct variable groups:   x, y, A   x, B, y   y, C   x, D

Allowed substitution hints:   C( x)   D( y)

Proof of Theorem iunxdif2

Step	Hyp	Ref	Expression
1		iunss2 4360	. . 3 |- ( A. x e. A E. y e. ( A \ B ) C C_ D -> U_ x e. A C C_ U_ y e. ( A \ B ) D )
2		difss 3617	. . . . 5 |- ( A \ B ) C_ A
3		iunss1 4327	. . . . 5 |- ( ( A \ B ) C_ A -> U_ y e. ( A \ B ) D C_ U_ y e. A D )
4	2, 3	ax-mp 5	. . . 4 |- U_ y e. ( A \ B ) D C_ U_ y e. A D
5		iunxdif2.1	. . . . 5 |- ( x = y -> C = D )
6	5	cbviunv 4354	. . . 4 |- U_ x e. A C = U_ y e. A D
7	4, 6	sseqtr4i 3522	. . 3 |- U_ y e. ( A \ B ) D C_ U_ x e. A C
8	1, 7	jctil 535	. 2 |- ( A. x e. A E. y e. ( A \ B ) C C_ D -> ( U_ y e. ( A \ B ) D C_ U_ x e. A C /\ U_ x e. A C C_ U_ y e. ( A \ B ) D ) )
9		eqss 3504	. 2 |- ( U_ y e. ( A \ B ) D = U_ x e. A C <-> ( U_ y e. ( A \ B ) D C_ U_ x e. A C /\ U_ x e. A C C_ U_ y e. ( A \ B ) D ) )
10	8, 9	sylibr 212	1 |- ( A. x e. A E. y e. ( A \ B ) C C_ D -> U_ y e. ( A \ B ) D = U_ x e. A C )

Syntax hints:   -> wi 4   /\ wa 367   = wceq 1398   A.wral 2804   E.wrex 2805   \ cdif 3458   C_ wss 3461   U_ciun 4315

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432

This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ral 2809  df-rex 2810  df-v 3108  df-dif 3464  df-in 3468  df-ss 3475  df-iun 4317

This theorem is referenced by: (None)