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Theorem iinss1 3764
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 3100 . . 3  |-  ( A 
C_  B  ->  ( A. x  e.  B  y  e.  C  ->  A. x  e.  A  y  e.  C ) )
2 vex 2636 . . . 4  |-  y  e. 
_V
3 eliin 3757 . . . 4  |-  ( y  e.  _V  ->  (
y  e.  |^|_ x  e.  B  C  <->  A. x  e.  B  y  e.  C ) )
42, 3ax-mp 7 . . 3  |-  ( y  e.  |^|_ x  e.  B  C 
<-> 
A. x  e.  B  y  e.  C )
5 eliin 3757 . . . 4  |-  ( y  e.  _V  ->  (
y  e.  |^|_ x  e.  A  C  <->  A. x  e.  A  y  e.  C ) )
62, 5ax-mp 7 . . 3  |-  ( y  e.  |^|_ x  e.  A  C 
<-> 
A. x  e.  A  y  e.  C )
71, 4, 63imtr4g 204 . 2  |-  ( A 
C_  B  ->  (
y  e.  |^|_ x  e.  B  C  ->  y  e.  |^|_ x  e.  A  C ) )
87ssrdv 3045 1  |-  ( A 
C_  B  ->  |^|_ x  e.  B  C  C_  |^|_ x  e.  A  C )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 104    e. wcel 1445   A.wral 2370   _Vcvv 2633    C_ wss 3013   |^|_ciin 3753
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 668  ax-5 1388  ax-7 1389  ax-gen 1390  ax-ie1 1434  ax-ie2 1435  ax-8 1447  ax-10 1448  ax-11 1449  ax-i12 1450  ax-bndl 1451  ax-4 1452  ax-17 1471  ax-i9 1475  ax-ial 1479  ax-i5r 1480  ax-ext 2077
This theorem depends on definitions:  df-bi 116  df-tru 1299  df-nf 1402  df-sb 1700  df-clab 2082  df-cleq 2088  df-clel 2091  df-nfc 2224  df-ral 2375  df-v 2635  df-in 3019  df-ss 3026  df-iin 3755
This theorem is referenced by: (None)
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