Theorem ssiun 3983

Description: Subset implication for an indexed union. (Contributed by NM, 3-Sep-2003.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)

Assertion

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

Distinct variable group:   x, C

Allowed substitution hints:   A( x)   B( x)

Proof of Theorem ssiun

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ssel 3195	. . . . 5 |- ( C C_ B -> ( y e. C -> y e. B ) )
2	1	reximi 2605	. . . 4 |- ( E. x e. A C C_ B -> E. x e. A ( y e. C -> y e. B ) )
3		r19.37av 2661	. . . 4 |- ( E. x e. A ( y e. C -> y e. B ) -> ( y e. C -> E. x e. A y e. B ) )
4	2, 3	syl 14	. . 3 |- ( E. x e. A C C_ B -> ( y e. C -> E. x e. A y e. B ) )
5		eliun 3945	. . 3 |- ( y e. U_ x e. A B <-> E. x e. A y e. B )
6	4, 5	imbitrrdi 162	. 2 |- ( E. x e. A C C_ B -> ( y e. C -> y e. U_ x e. A B ) )
7	6	ssrdv 3207	1 |- ( E. x e. A C C_ B -> C C_ U_ x e. A B )

Syntax hints:   -> wi 4   e. wcel 2178   E.wrex 2487   C_ wss 3174   U_ciun 3941

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2189

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ral 2491  df-rex 2492  df-v 2778  df-in 3180  df-ss 3187  df-iun 3943

This theorem is referenced by:  iunss2  3986  iunpwss  4033  iunpw  4545