Theorem iunxdif2 4568

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 4565	. . 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 3737	. . . . 5 |- ( A \ B ) C_ A
3		iunss1 4532	. . . . 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 4559	. . . 4 |- U_ x e. A C = U_ y e. A D
7	4, 6	sseqtr4i 3638	. . 3 |- U_ y e. ( A \ B ) D C_ U_ x e. A C
8	1, 7	jctil 560	. 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 3618	. 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 224	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 384   = wceq 1483   A.wral 2912   E.wrex 2913   \ cdif 3571   C_ wss 3574   U_ciun 4520

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-v 3202  df-dif 3577  df-in 3581  df-ss 3588  df-iun 4522

This theorem is referenced by: (None)