Theorem iunxdif2 3726

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 3723	. . 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 3098	. . . . 5 |- ( A \ B ) C_ A
3		iunss1 3689	. . . . 5 |- ( ( A \ B ) C_ A -> U_ y e. ( A \ B ) D C_ U_ y e. A D )
4	2, 3	ax-mp 7	. . . 4 |- U_ y e. ( A \ B ) D C_ U_ y e. A D
5		iunxdif2.1	. . . . 5 |- ( x = y -> C = D )
6	5	cbviunv 3717	. . . 4 |- U_ x e. A C = U_ y e. A D
7	4, 6	sseqtr4i 3032	. . 3 |- U_ y e. ( A \ B ) D C_ U_ x e. A C
8	1, 7	jctil 305	. 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 3014	. 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 132	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 102   = wceq 1284   A.wral 2348   E.wrex 2349   \ cdif 2970   C_ wss 2973   U_ciun 3678

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 576  ax-in2 577  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ral 2353  df-rex 2354  df-v 2603  df-dif 2975  df-in 2979  df-ss 2986  df-iun 3680

This theorem is referenced by: (None)