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Theorem fnopabg 5053
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 2017 . . . . . 6  |-  ( E* y ( x  e.  A  /\  ph )  <->  ( x  e.  A  ->  E* y ph ) )
21albii 1400 . . . . 5  |-  ( A. x E* y ( x  e.  A  /\  ph ) 
<-> 
A. x ( x  e.  A  ->  E* y ph ) )
3 funopab 4965 . . . . 5  |-  ( Fun 
{ <. x ,  y
>.  |  ( x  e.  A  /\  ph ) } 
<-> 
A. x E* y
( x  e.  A  /\  ph ) )
4 df-ral 2354 . . . . 5  |-  ( A. x  e.  A  E* y ph  <->  A. x ( x  e.  A  ->  E* y ph ) )
52, 3, 43bitr4ri 211 . . . 4  |-  ( A. x  e.  A  E* y ph  <->  Fun  { <. x ,  y >.  |  ( x  e.  A  /\  ph ) } )
6 dmopab3 4576 . . . 4  |-  ( A. x  e.  A  E. y ph  <->  dom  { <. x ,  y >.  |  ( x  e.  A  /\  ph ) }  =  A )
75, 6anbi12i 448 . . 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 2486 . . 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 4935 . . 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 210 . 2  |-  ( A. x  e.  A  ( E* y ph  /\  E. y ph )  <->  { <. x ,  y >.  |  ( x  e.  A  /\  ph ) }  Fn  A
)
11 eu5 1989 . . . 4  |-  ( E! y ph  <->  ( E. y ph  /\  E* y ph ) )
12 ancom 262 . . . 4  |-  ( ( E. y ph  /\  E* y ph )  <->  ( E* y ph  /\  E. y ph ) )
1311, 12bitri 182 . . 3  |-  ( E! y ph  <->  ( E* y ph  /\  E. y ph ) )
1413ralbii 2373 . 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 5024 . 2  |-  ( F  Fn  A  <->  { <. x ,  y >.  |  ( x  e.  A  /\  ph ) }  Fn  A
)
1710, 14, 163bitr4i 210 1  |-  ( A. x  e.  A  E! y ph  <->  F  Fn  A
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    <-> wb 103   A.wal 1283    = wceq 1285   E.wex 1422    e. wcel 1434   E!weu 1942   E*wmo 1943   A.wral 2349   {copab 3846   dom cdm 4371   Fun wfun 4926    Fn wfn 4927
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 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-sep 3904  ax-pow 3956  ax-pr 3972
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1687  df-eu 1945  df-mo 1946  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ral 2354  df-rex 2355  df-v 2604  df-un 2978  df-in 2980  df-ss 2987  df-pw 3392  df-sn 3412  df-pr 3413  df-op 3415  df-br 3794  df-opab 3848  df-id 4056  df-xp 4377  df-rel 4378  df-cnv 4379  df-co 4380  df-dm 4381  df-fun 4934  df-fn 4935
This theorem is referenced by:  fnopab  5054  mptfng  5055
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