Theorem iunss2 4015

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 3924. (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 4012	. . 3 |- ( E. y e. B C C_ D -> C C_ U_ y e. B D )
2	1	ralimi 2595	. 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 4011	. 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 2510   E.wrex 2511   C_ wss 3200   U_ciun 3970

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-rex 2516  df-v 2804  df-in 3206  df-ss 3213  df-iun 3972

This theorem is referenced by:  iunxdif2  4019  rdgss  6548