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Theorem iinss1 3913
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 3234 . . 3  |-  ( A 
C_  B  ->  ( A. x  e.  B  y  e.  C  ->  A. x  e.  A  y  e.  C ) )
2 vex 2755 . . . 4  |-  y  e. 
_V
3 eliin 3906 . . . 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 3906 . . . 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 205 . 2  |-  ( A 
C_  B  ->  (
y  e.  |^|_ x  e.  B  C  ->  y  e.  |^|_ x  e.  A  C ) )
87ssrdv 3176 1  |-  ( A 
C_  B  ->  |^|_ x  e.  B  C  C_  |^|_ x  e.  A  C )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 105    e. wcel 2160   A.wral 2468   _Vcvv 2752    C_ wss 3144   |^|_ciin 3902
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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171
This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-v 2754  df-in 3150  df-ss 3157  df-iin 3904
This theorem is referenced by: (None)
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