Theorem iinss2 4572

Description: An indexed intersection is included in any of its members. (Contributed by FL, 15-Oct-2012.)

Assertion

Ref	Expression
iinss2	|- ( x e. A -> |^|_ x e. A B C_ B )

Proof of Theorem iinss2

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		vex 3203	. . . 4 |- y e. _V
2		eliin 4525	. . . 4 |- ( y e. _V -> ( y e. |^|_ x e. A B <-> A. x e. A y e. B ) )
3	1, 2	ax-mp 5	. . 3 |- ( y e. |^|_ x e. A B <-> A. x e. A y e. B )
4		rsp 2929	. . . 4 |- ( A. x e. A y e. B -> ( x e. A -> y e. B ) )
5	4	com12 32	. . 3 |- ( x e. A -> ( A. x e. A y e. B -> y e. B ) )
6	3, 5	syl5bi 232	. 2 |- ( x e. A -> ( y e. |^|_ x e. A B -> y e. B ) )
7	6	ssrdv 3609	1 |- ( x e. A -> |^|_ x e. A B C_ B )

Syntax hints:   -> wi 4   <-> wb 196   e. wcel 1990   A.wral 2912   _Vcvv 3200   C_ wss 3574   |^|_ciin 4521

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-v 3202  df-in 3581  df-ss 3588  df-iin 4523

This theorem is referenced by:  dmiin  5369  gruiin  9632  txtube  21443  iooiinicc  39769  iooiinioc  39783  meaiininclem  40700  smfsuplem1  41017  smfsuplem3  41019  smflimsuplem2  41027