Theorem ssiun2 3959

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 2546	. . . 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 3920	. . 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 3189	1 |- ( x e. A -> B C_ U_ x e. A B )

Syntax hints:   -> wi 4   e. wcel 2167   E.wrex 2476   C_ wss 3157   U_ciun 3916

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 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-rex 2481  df-v 2765  df-in 3163  df-ss 3170  df-iun 3918

This theorem is referenced by:  ssiun2s  3960  triun  4144  ixpf  6779