Theorem iunss2 3894

Description: A subclass condition on the members of two indexed classes C ( x ) and D ( y ) that implies a subclass relation on their indexed unions. Generalization of Proposition 8.6 of [TakeutiZaring] p. 59. Compare uniss2 3803. (Contributed by NM, 9-Dec-2004.)

Assertion

Ref	Expression
iunss2	|- ( A. x e. A E. y e. B C C_ D -> U_ x e. A C C_ U_ y e. B D )

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

Allowed substitution hints:   A( x, y)   B( y)   C( x)   D( y)

Proof of Theorem iunss2

Step	Hyp	Ref	Expression
1		ssiun 3891	. . 3 |- ( E. y e. B C C_ D -> C C_ U_ y e. B D )
2	1	ralimi 2520	. 2 |- ( A. x e. A E. y e. B C C_ D -> A. x e. A C C_ U_ y e. B D )
3		iunss 3890	. 2 |- ( U_ x e. A C C_ U_ y e. B D <-> A. x e. A C C_ U_ y e. B D )
4	2, 3	sylibr 133	1 |- ( A. x e. A E. y e. B C C_ D -> U_ x e. A C C_ U_ y e. B D )

Syntax hints:   -> wi 4   A.wral 2435   E.wrex 2436   C_ wss 3102   U_ciun 3849

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2139

This theorem depends on definitions:  df-bi 116  df-tru 1338  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ral 2440  df-rex 2441  df-v 2714  df-in 3108  df-ss 3115  df-iun 3851

This theorem is referenced by:  iunxdif2  3897  rdgss  6330