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Theorem ssiinf 4141
Description: Subset theorem for an indexed intersection. (Contributed by FL, 15-Oct-2012.) (Proof shortened by Mario Carneiro, 14-Oct-2016.)
Hypothesis
Ref Expression
ssiinf.1  |-  F/_ x C
Assertion
Ref Expression
ssiinf  |-  ( C 
C_  |^|_ x  e.  A  B 
<-> 
A. x  e.  A  C  C_  B )

Proof of Theorem ssiinf
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 vex 2960 . . . . 5  |-  y  e. 
_V
2 eliin 4099 . . . . 5  |-  ( y  e.  _V  ->  (
y  e.  |^|_ x  e.  A  B  <->  A. x  e.  A  y  e.  B ) )
31, 2ax-mp 8 . . . 4  |-  ( y  e.  |^|_ x  e.  A  B 
<-> 
A. x  e.  A  y  e.  B )
43ralbii 2730 . . 3  |-  ( A. y  e.  C  y  e.  |^|_ x  e.  A  B 
<-> 
A. y  e.  C  A. x  e.  A  y  e.  B )
5 ssiinf.1 . . . 4  |-  F/_ x C
6 nfcv 2573 . . . 4  |-  F/_ y A
75, 6ralcomf 2867 . . 3  |-  ( A. y  e.  C  A. x  e.  A  y  e.  B  <->  A. x  e.  A  A. y  e.  C  y  e.  B )
84, 7bitri 242 . 2  |-  ( A. y  e.  C  y  e.  |^|_ x  e.  A  B 
<-> 
A. x  e.  A  A. y  e.  C  y  e.  B )
9 dfss3 3339 . 2  |-  ( C 
C_  |^|_ x  e.  A  B 
<-> 
A. y  e.  C  y  e.  |^|_ x  e.  A  B )
10 dfss3 3339 . . 3  |-  ( C 
C_  B  <->  A. y  e.  C  y  e.  B )
1110ralbii 2730 . 2  |-  ( A. x  e.  A  C  C_  B  <->  A. x  e.  A  A. y  e.  C  y  e.  B )
128, 9, 113bitr4i 270 1  |-  ( C 
C_  |^|_ x  e.  A  B 
<-> 
A. x  e.  A  C  C_  B )
Colors of variables: wff set class
Syntax hints:    <-> wb 178    e. wcel 1726   F/_wnfc 2560   A.wral 2706   _Vcvv 2957    C_ wss 3321   |^|_ciin 4095
This theorem is referenced by:  ssiin  4142  dmiin  5114
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2418
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ral 2711  df-v 2959  df-in 3328  df-ss 3335  df-iin 4097
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