Theorem iunss2 4052

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 3961. (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 4049	. . 3 |- ( E. y e. B C C_ D -> C C_ U_ y e. B D )
2	1	ralimi 2613	. 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 4048	. 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 2528   E.wrex 2529   C_ wss 3220   U_ciun 4007

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-v 2823  df-in 3226  df-ss 3233  df-iun 4009

This theorem is referenced by:  iunxdif2  4056  rdgss  6644