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Theorem abnexg 4446
Description: Sufficient condition for a class abstraction to be a proper class. The class  F can be thought of as an expression in  x and the abstraction appearing in the statement as the class of values  F as  x varies through  A. Assuming the antecedents, if that class is a set, then so is the "domain"  A. The converse holds without antecedent, see abrexexg 6118. Note that the second antecedent  A. x  e.  A x  e.  F cannot be translated to  A  C_  F since  F may depend on  x. In applications, one may take  F  =  { x } or  F  =  ~P x (see snnex 4448 and pwnex 4449 respectively, proved from abnex 4447, which is a consequence of abnexg 4446 with  A  =  _V). (Contributed by BJ, 2-Dec-2021.)
Assertion
Ref Expression
abnexg  |-  ( A. x  e.  A  ( F  e.  V  /\  x  e.  F )  ->  ( { y  |  E. x  e.  A  y  =  F }  e.  W  ->  A  e. 
_V ) )
Distinct variable groups:    x, A, y   
y, F
Allowed substitution hints:    F( x)    V( x, y)    W( x, y)

Proof of Theorem abnexg
StepHypRef Expression
1 uniexg 4439 . 2  |-  ( { y  |  E. x  e.  A  y  =  F }  e.  W  ->  U. { y  |  E. x  e.  A  y  =  F }  e.  _V )
2 simpl 109 . . . . 5  |-  ( ( F  e.  V  /\  x  e.  F )  ->  F  e.  V )
32ralimi 2540 . . . 4  |-  ( A. x  e.  A  ( F  e.  V  /\  x  e.  F )  ->  A. x  e.  A  F  e.  V )
4 dfiun2g 3918 . . . . . 6  |-  ( A. x  e.  A  F  e.  V  ->  U_ x  e.  A  F  =  U. { y  |  E. x  e.  A  y  =  F } )
54eleq1d 2246 . . . . 5  |-  ( A. x  e.  A  F  e.  V  ->  ( U_ x  e.  A  F  e.  _V  <->  U. { y  |  E. x  e.  A  y  =  F }  e.  _V ) )
65biimprd 158 . . . 4  |-  ( A. x  e.  A  F  e.  V  ->  ( U. { y  |  E. x  e.  A  y  =  F }  e.  _V  ->  U_ x  e.  A  F  e.  _V )
)
73, 6syl 14 . . 3  |-  ( A. x  e.  A  ( F  e.  V  /\  x  e.  F )  ->  ( U. { y  |  E. x  e.  A  y  =  F }  e.  _V  ->  U_ x  e.  A  F  e.  _V ) )
8 simpr 110 . . . . 5  |-  ( ( F  e.  V  /\  x  e.  F )  ->  x  e.  F )
98ralimi 2540 . . . 4  |-  ( A. x  e.  A  ( F  e.  V  /\  x  e.  F )  ->  A. x  e.  A  x  e.  F )
10 iunid 3942 . . . . 5  |-  U_ x  e.  A  { x }  =  A
11 snssi 3736 . . . . . . 7  |-  ( x  e.  F  ->  { x }  C_  F )
1211ralimi 2540 . . . . . 6  |-  ( A. x  e.  A  x  e.  F  ->  A. x  e.  A  { x }  C_  F )
13 ss2iun 3901 . . . . . 6  |-  ( A. x  e.  A  {
x }  C_  F  ->  U_ x  e.  A  { x }  C_  U_ x  e.  A  F
)
1412, 13syl 14 . . . . 5  |-  ( A. x  e.  A  x  e.  F  ->  U_ x  e.  A  { x }  C_  U_ x  e.  A  F )
1510, 14eqsstrrid 3202 . . . 4  |-  ( A. x  e.  A  x  e.  F  ->  A  C_  U_ x  e.  A  F
)
16 ssexg 4142 . . . . 5  |-  ( ( A  C_  U_ x  e.  A  F  /\  U_ x  e.  A  F  e.  _V )  ->  A  e.  _V )
1716ex 115 . . . 4  |-  ( A 
C_  U_ x  e.  A  F  ->  ( U_ x  e.  A  F  e.  _V  ->  A  e.  _V ) )
189, 15, 173syl 17 . . 3  |-  ( A. x  e.  A  ( F  e.  V  /\  x  e.  F )  ->  ( U_ x  e.  A  F  e.  _V  ->  A  e.  _V )
)
197, 18syld 45 . 2  |-  ( A. x  e.  A  ( F  e.  V  /\  x  e.  F )  ->  ( U. { y  |  E. x  e.  A  y  =  F }  e.  _V  ->  A  e.  _V ) )
201, 19syl5 32 1  |-  ( A. x  e.  A  ( F  e.  V  /\  x  e.  F )  ->  ( { y  |  E. x  e.  A  y  =  F }  e.  W  ->  A  e. 
_V ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    = wceq 1353    e. wcel 2148   {cab 2163   A.wral 2455   E.wrex 2456   _Vcvv 2737    C_ wss 3129   {csn 3592   U.cuni 3809   U_ciun 3886
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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4121  ax-un 4433
This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2739  df-in 3135  df-ss 3142  df-sn 3598  df-uni 3810  df-iun 3888
This theorem is referenced by:  abnex  4447
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