Theorem iunss2 3918

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 3827. (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 3915	. . 3 |- ( E. y e. B C C_ D -> C C_ U_ y e. B D )
2	1	ralimi 2533	. 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 3914	. 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 2448   E.wrex 2449   C_ wss 3121   U_ciun 3873

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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-in 3127  df-ss 3134  df-iun 3875

This theorem is referenced by:  iunxdif2  3921  rdgss  6362