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Theorem fnopabg 5339
Description: Functionality and domain of an ordered-pair class abstraction. (Contributed by NM, 30-Jan-2004.) (Proof shortened by Mario Carneiro, 4-Dec-2016.)
Hypothesis
Ref Expression
fnopabg.1  |-  F  =  { <. x ,  y
>.  |  ( x  e.  A  /\  ph ) }
Assertion
Ref Expression
fnopabg  |-  ( A. x  e.  A  E! y ph  <->  F  Fn  A
)
Distinct variable group:    x, y, A
Allowed substitution hints:    ph( x, y)    F( x, y)

Proof of Theorem fnopabg
StepHypRef Expression
1 moanimv 2101 . . . . . 6  |-  ( E* y ( x  e.  A  /\  ph )  <->  ( x  e.  A  ->  E* y ph ) )
21albii 1470 . . . . 5  |-  ( A. x E* y ( x  e.  A  /\  ph ) 
<-> 
A. x ( x  e.  A  ->  E* y ph ) )
3 funopab 5251 . . . . 5  |-  ( Fun 
{ <. x ,  y
>.  |  ( x  e.  A  /\  ph ) } 
<-> 
A. x E* y
( x  e.  A  /\  ph ) )
4 df-ral 2460 . . . . 5  |-  ( A. x  e.  A  E* y ph  <->  A. x ( x  e.  A  ->  E* y ph ) )
52, 3, 43bitr4ri 213 . . . 4  |-  ( A. x  e.  A  E* y ph  <->  Fun  { <. x ,  y >.  |  ( x  e.  A  /\  ph ) } )
6 dmopab3 4840 . . . 4  |-  ( A. x  e.  A  E. y ph  <->  dom  { <. x ,  y >.  |  ( x  e.  A  /\  ph ) }  =  A )
75, 6anbi12i 460 . . 3  |-  ( ( A. x  e.  A  E* y ph  /\  A. x  e.  A  E. y ph )  <->  ( Fun  {
<. x ,  y >.  |  ( x  e.  A  /\  ph ) }  /\  dom  { <. x ,  y >.  |  ( x  e.  A  /\  ph ) }  =  A ) )
8 r19.26 2603 . . 3  |-  ( A. x  e.  A  ( E* y ph  /\  E. y ph )  <->  ( A. x  e.  A  E* y ph  /\  A. x  e.  A  E. y ph ) )
9 df-fn 5219 . . 3  |-  ( {
<. x ,  y >.  |  ( x  e.  A  /\  ph ) }  Fn  A  <->  ( Fun  {
<. x ,  y >.  |  ( x  e.  A  /\  ph ) }  /\  dom  { <. x ,  y >.  |  ( x  e.  A  /\  ph ) }  =  A ) )
107, 8, 93bitr4i 212 . 2  |-  ( A. x  e.  A  ( E* y ph  /\  E. y ph )  <->  { <. x ,  y >.  |  ( x  e.  A  /\  ph ) }  Fn  A
)
11 eu5 2073 . . . 4  |-  ( E! y ph  <->  ( E. y ph  /\  E* y ph ) )
12 ancom 266 . . . 4  |-  ( ( E. y ph  /\  E* y ph )  <->  ( E* y ph  /\  E. y ph ) )
1311, 12bitri 184 . . 3  |-  ( E! y ph  <->  ( E* y ph  /\  E. y ph ) )
1413ralbii 2483 . 2  |-  ( A. x  e.  A  E! y ph  <->  A. x  e.  A  ( E* y ph  /\  E. y ph ) )
15 fnopabg.1 . . 3  |-  F  =  { <. x ,  y
>.  |  ( x  e.  A  /\  ph ) }
1615fneq1i 5310 . 2  |-  ( F  Fn  A  <->  { <. x ,  y >.  |  ( x  e.  A  /\  ph ) }  Fn  A
)
1710, 14, 163bitr4i 212 1  |-  ( A. x  e.  A  E! y ph  <->  F  Fn  A
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    <-> wb 105   A.wal 1351    = wceq 1353   E.wex 1492   E!weu 2026   E*wmo 2027    e. wcel 2148   A.wral 2455   {copab 4063   dom cdm 4626   Fun wfun 5210    Fn wfn 5211
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-14 2151  ax-ext 2159  ax-sep 4121  ax-pow 4174  ax-pr 4209
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2739  df-un 3133  df-in 3135  df-ss 3142  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-br 4004  df-opab 4065  df-id 4293  df-xp 4632  df-rel 4633  df-cnv 4634  df-co 4635  df-dm 4636  df-fun 5218  df-fn 5219
This theorem is referenced by:  fnopab  5340  mptfng  5341
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