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Theorem iinss1 4283
Description: Subclass theorem for indexed intersection. (Contributed by NM, 24-Jan-2012.)
Assertion
Ref Expression
iinss1  |-  ( A 
C_  B  ->  |^|_ x  e.  B  C  C_  |^|_ x  e.  A  C )
Distinct variable groups:    x, A    x, B
Allowed substitution hint:    C( x)

Proof of Theorem iinss1
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 ssralv 3502 . . 3  |-  ( A 
C_  B  ->  ( A. x  e.  B  y  e.  C  ->  A. x  e.  A  y  e.  C ) )
2 vex 3061 . . . 4  |-  y  e. 
_V
3 eliin 4276 . . . 4  |-  ( y  e.  _V  ->  (
y  e.  |^|_ x  e.  B  C  <->  A. x  e.  B  y  e.  C ) )
42, 3ax-mp 5 . . 3  |-  ( y  e.  |^|_ x  e.  B  C 
<-> 
A. x  e.  B  y  e.  C )
5 eliin 4276 . . . 4  |-  ( y  e.  _V  ->  (
y  e.  |^|_ x  e.  A  C  <->  A. x  e.  A  y  e.  C ) )
62, 5ax-mp 5 . . 3  |-  ( y  e.  |^|_ x  e.  A  C 
<-> 
A. x  e.  A  y  e.  C )
71, 4, 63imtr4g 270 . 2  |-  ( A 
C_  B  ->  (
y  e.  |^|_ x  e.  B  C  ->  y  e.  |^|_ x  e.  A  C ) )
87ssrdv 3447 1  |-  ( A 
C_  B  ->  |^|_ x  e.  B  C  C_  |^|_ x  e.  A  C )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    e. wcel 1842   A.wral 2753   _Vcvv 3058    C_ wss 3413   |^|_ciin 4271
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380
This theorem depends on definitions:  df-bi 185  df-an 369  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ral 2758  df-v 3060  df-in 3420  df-ss 3427  df-iin 4273
This theorem is referenced by:  polcon3N  32914
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