Theorem ss2iun 4022

Description: Subclass theorem for indexed union. (Contributed by NM, 26-Nov-2003.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)

Assertion

Ref	Expression
ss2iun	|- ( A. x e. A B C_ C -> U_ x e. A B C_ U_ x e. A C )

Proof of Theorem ss2iun

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ssel 3242	. . . . 5 |- ( B C_ C -> ( y e. B -> y e. C ) )
2	1	ralimi 2613	. . . 4 |- ( A. x e. A B C_ C -> A. x e. A ( y e. B -> y e. C ) )
3		rexim 2644	. . . 4 |- ( A. x e. A ( y e. B -> y e. C ) -> ( E. x e. A y e. B -> E. x e. A y e. C ) )
4	2, 3	syl 14	. . 3 |- ( A. x e. A B C_ C -> ( E. x e. A y e. B -> E. x e. A y e. C ) )
5		eliun 4011	. . 3 |- ( y e. U_ x e. A B <-> E. x e. A y e. B )
6		eliun 4011	. . 3 |- ( y e. U_ x e. A C <-> E. x e. A y e. C )
7	4, 5, 6	3imtr4g 205	. 2 |- ( A. x e. A B C_ C -> ( y e. U_ x e. A B -> y e. U_ x e. A C ) )
8	7	ssrdv 3254	1 |- ( A. x e. A B C_ C -> U_ x e. A B C_ U_ x e. A C )

Syntax hints:   -> wi 4   e. wcel 2209   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:  iuneq2  4023  abnexg  4587  imasaddvallemg  13613  dvfvalap  15705