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Theorem iinss1 3998
Description: Subclass theorem for indexed union. (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 3313 . . 3  |-  ( A 
C_  B  ->  ( A. x  e.  B  y  e.  C  ->  A. x  e.  A  y  e.  C ) )
2 vex 2867 . . . 4  |-  y  e. 
_V
3 eliin 3991 . . . 4  |-  ( y  e.  _V  ->  (
y  e.  |^|_ x  e.  B  C  <->  A. x  e.  B  y  e.  C ) )
42, 3ax-mp 8 . . 3  |-  ( y  e.  |^|_ x  e.  B  C 
<-> 
A. x  e.  B  y  e.  C )
5 eliin 3991 . . . 4  |-  ( y  e.  _V  ->  (
y  e.  |^|_ x  e.  A  C  <->  A. x  e.  A  y  e.  C ) )
62, 5ax-mp 8 . . 3  |-  ( y  e.  |^|_ x  e.  A  C 
<-> 
A. x  e.  A  y  e.  C )
71, 4, 63imtr4g 261 . 2  |-  ( A 
C_  B  ->  (
y  e.  |^|_ x  e.  B  C  ->  y  e.  |^|_ x  e.  A  C ) )
87ssrdv 3261 1  |-  ( A 
C_  B  ->  |^|_ x  e.  B  C  C_  |^|_ x  e.  A  C )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    e. wcel 1710   A.wral 2619   _Vcvv 2864    C_ wss 3228   |^|_ciin 3987
This theorem is referenced by:  polcon3N  30175
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ral 2624  df-v 2866  df-in 3235  df-ss 3242  df-iin 3989
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