Theorem ssiin 4570

Description: Subset theorem for an indexed intersection. (Contributed by NM, 15-Oct-2003.)

Assertion

Ref	Expression
ssiin	|- ( C C_ |^|_ x e. A B <-> A. x e. A C C_ B )

Distinct variable group:   x, C

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

Proof of Theorem ssiin

Step	Hyp	Ref	Expression
1		nfcv 2764	. 2 |- F/_ x C
2	1	ssiinf 4569	1 |- ( C C_ |^|_ x e. A B <-> A. x e. A C C_ B )

Syntax hints:   <-> wb 196   A.wral 2912   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:  cflim2  9085  ptbasfi  21384  limciun  23658  clsint2  32324  fnemeet2  32362  dihglblem4  36586  dihglblem6  36629  iooiinicc  39769  iooiinioc  39783  iinhoiicc  40888  smfsuplem1  41017