Theorem iunss2 3933

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 3842. (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 3930	. . 3 |- ( E. y e. B C C_ D -> C C_ U_ y e. B D )
2	1	ralimi 2540	. 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 3929	. 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 134	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 2455   E.wrex 2456   C_ wss 3131   U_ciun 3888

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-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2741  df-in 3137  df-ss 3144  df-iun 3890

This theorem is referenced by:  iunxdif2  3937  rdgss  6386