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Theorem ssiinf 3779
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 2622 . . . . 5  |-  y  e. 
_V
2 eliin 3735 . . . . 5  |-  ( y  e.  _V  ->  (
y  e.  |^|_ x  e.  A  B  <->  A. x  e.  A  y  e.  B ) )
31, 2ax-mp 7 . . . 4  |-  ( y  e.  |^|_ x  e.  A  B 
<-> 
A. x  e.  A  y  e.  B )
43ralbii 2384 . . 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 2228 . . . 4  |-  F/_ y A
75, 6ralcomf 2528 . . 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 182 . 2  |-  ( A. y  e.  C  y  e.  |^|_ x  e.  A  B 
<-> 
A. x  e.  A  A. y  e.  C  y  e.  B )
9 dfss3 3015 . 2  |-  ( C 
C_  |^|_ x  e.  A  B 
<-> 
A. y  e.  C  y  e.  |^|_ x  e.  A  B )
10 dfss3 3015 . . 3  |-  ( C 
C_  B  <->  A. y  e.  C  y  e.  B )
1110ralbii 2384 . 2  |-  ( A. x  e.  A  C  C_  B  <->  A. x  e.  A  A. y  e.  C  y  e.  B )
128, 9, 113bitr4i 210 1  |-  ( C 
C_  |^|_ x  e.  A  B 
<-> 
A. x  e.  A  C  C_  B )
Colors of variables: wff set class
Syntax hints:    <-> wb 103    e. wcel 1438   F/_wnfc 2215   A.wral 2359   _Vcvv 2619    C_ wss 2999   |^|_ciin 3731
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-v 2621  df-in 3005  df-ss 3012  df-iin 3733
This theorem is referenced by:  ssiin  3780  dmiin  4681
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