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Theorem abnexg 4431
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 6097. 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 4433 and pwnex 4434 respectively, proved from abnex 4432, which is a consequence of abnexg 4431 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 4424 . 2  |-  ( { y  |  E. x  e.  A  y  =  F }  e.  W  ->  U. { y  |  E. x  e.  A  y  =  F }  e.  _V )
2 simpl 108 . . . . 5  |-  ( ( F  e.  V  /\  x  e.  F )  ->  F  e.  V )
32ralimi 2533 . . . 4  |-  ( A. x  e.  A  ( F  e.  V  /\  x  e.  F )  ->  A. x  e.  A  F  e.  V )
4 dfiun2g 3905 . . . . . 6  |-  ( A. x  e.  A  F  e.  V  ->  U_ x  e.  A  F  =  U. { y  |  E. x  e.  A  y  =  F } )
54eleq1d 2239 . . . . 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 157 . . . 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 109 . . . . 5  |-  ( ( F  e.  V  /\  x  e.  F )  ->  x  e.  F )
98ralimi 2533 . . . 4  |-  ( A. x  e.  A  ( F  e.  V  /\  x  e.  F )  ->  A. x  e.  A  x  e.  F )
10 iunid 3928 . . . . 5  |-  U_ x  e.  A  { x }  =  A
11 snssi 3724 . . . . . . 7  |-  ( x  e.  F  ->  { x }  C_  F )
1211ralimi 2533 . . . . . 6  |-  ( A. x  e.  A  x  e.  F  ->  A. x  e.  A  { x }  C_  F )
13 ss2iun 3888 . . . . . 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 3194 . . . 4  |-  ( A. x  e.  A  x  e.  F  ->  A  C_  U_ x  e.  A  F
)
16 ssexg 4128 . . . . 5  |-  ( ( A  C_  U_ x  e.  A  F  /\  U_ x  e.  A  F  e.  _V )  ->  A  e.  _V )
1716ex 114 . . . 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 103    = wceq 1348    e. wcel 2141   {cab 2156   A.wral 2448   E.wrex 2449   _Vcvv 2730    C_ wss 3121   {csn 3583   U.cuni 3796   U_ciun 3873
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-un 4418
This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-in 3127  df-ss 3134  df-sn 3589  df-uni 3797  df-iun 3875
This theorem is referenced by:  abnex  4432
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