Theorem ssiun2 3929

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 2526	. . . 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 3890	. . 3 |- ( y e. U_ x e. A B <-> E. x e. A y e. B )
4	2, 3	syl6ibr 162	. 2 |- ( x e. A -> ( y e. B -> y e. U_ x e. A B ) )
5	4	ssrdv 3161	1 |- ( x e. A -> B C_ U_ x e. A B )

Syntax hints:   -> wi 4   e. wcel 2148   E.wrex 2456   C_ wss 3129   U_ciun 3886

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 2739  df-in 3135  df-ss 3142  df-iun 3888

This theorem is referenced by:  ssiun2s  3930  triun  4113  ixpf  6716