Theorem ssiun2 3779

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 2425	. . . 4 |- ( ( x e. A /\ y e. B ) -> E. x e. A y e. B )
2	1	ex 114	. . 3 |- ( x e. A -> ( y e. B -> E. x e. A y e. B ) )
3		eliun 3740	. . 3 |- ( y e. U_ x e. A B <-> E. x e. A y e. B )
4	2, 3	syl6ibr 161	. 2 |- ( x e. A -> ( y e. B -> y e. U_ x e. A B ) )
5	4	ssrdv 3032	1 |- ( x e. A -> B C_ U_ x e. A B )

Syntax hints:   -> wi 4   e. wcel 1439   E.wrex 2361   C_ wss 3000   U_ciun 3736

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071

This theorem depends on definitions:  df-bi 116  df-tru 1293  df-nf 1396  df-sb 1694  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-ral 2365  df-rex 2366  df-v 2622  df-in 3006  df-ss 3013  df-iun 3738

This theorem is referenced by:  ssiun2s  3780  triun  3955  ixpf  6491