Theorem iunxdif2 4024

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 4020	. . 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 3335	. . . . 5 |- ( A \ B ) C_ A
3		iunss1 3986	. . . . 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 4014	. . . 4 |- U_ x e. A C = U_ y e. A D
7	4, 6	sseqtrri 3263	. . 3 |- U_ y e. ( A \ B ) D C_ U_ x e. A C
8	1, 7	jctil 312	. 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 3243	. 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 134	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 104   = wceq 1398   A.wral 2511   E.wrex 2512   \ cdif 3198   C_ wss 3201   U_ciun 3975

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ral 2516  df-rex 2517  df-v 2805  df-dif 3203  df-in 3207  df-ss 3214  df-iun 3977

This theorem is referenced by: (None)