Theorem ssiun2 3987

Description: Identity law for subset of an indexed union. (Contributed by NM, 12-Oct-2003.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)

Assertion

Ref	Expression
ssiun2	|- ( x e. A -> B C_ U_ x e. A B )

Proof of Theorem ssiun2

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		rspe 2559	. . . 4 |- ( ( x e. A /\ y e. B ) -> E. x e. A y e. B )
2	1	ex 115	. . 3 |- ( x e. A -> ( y e. B -> E. x e. A y e. B ) )
3		eliun 3948	. . 3 |- ( y e. U_ x e. A B <-> E. x e. A y e. B )
4	2, 3	imbitrrdi 162	. 2 |- ( x e. A -> ( y e. B -> y e. U_ x e. A B ) )
5	4	ssrdv 3210	1 |- ( x e. A -> B C_ U_ x e. A B )

Syntax hints:   -> wi 4   e. wcel 2180   E.wrex 2489   C_ wss 3177   U_ciun 3944

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 713  ax-5 1473  ax-7 1474  ax-gen 1475  ax-ie1 1519  ax-ie2 1520  ax-8 1530  ax-10 1531  ax-11 1532  ax-i12 1533  ax-bndl 1535  ax-4 1536  ax-17 1552  ax-i9 1556  ax-ial 1560  ax-i5r 1561  ax-ext 2191

This theorem depends on definitions:  df-bi 117  df-tru 1378  df-nf 1487  df-sb 1789  df-clab 2196  df-cleq 2202  df-clel 2205  df-nfc 2341  df-ral 2493  df-rex 2494  df-v 2781  df-in 3183  df-ss 3190  df-iun 3946

This theorem is referenced by:  ssiun2s  3988  triun  4174  ixpf  6837