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Theorem fnopabg 5241
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 2072 . . . . . 6  |-  ( E* y ( x  e.  A  /\  ph )  <->  ( x  e.  A  ->  E* y ph ) )
21albii 1446 . . . . 5  |-  ( A. x E* y ( x  e.  A  /\  ph ) 
<-> 
A. x ( x  e.  A  ->  E* y ph ) )
3 funopab 5153 . . . . 5  |-  ( Fun 
{ <. x ,  y
>.  |  ( x  e.  A  /\  ph ) } 
<-> 
A. x E* y
( x  e.  A  /\  ph ) )
4 df-ral 2419 . . . . 5  |-  ( A. x  e.  A  E* y ph  <->  A. x ( x  e.  A  ->  E* y ph ) )
52, 3, 43bitr4ri 212 . . . 4  |-  ( A. x  e.  A  E* y ph  <->  Fun  { <. x ,  y >.  |  ( x  e.  A  /\  ph ) } )
6 dmopab3 4747 . . . 4  |-  ( A. x  e.  A  E. y ph  <->  dom  { <. x ,  y >.  |  ( x  e.  A  /\  ph ) }  =  A )
75, 6anbi12i 455 . . 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 2556 . . 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 5121 . . 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 211 . 2  |-  ( A. x  e.  A  ( E* y ph  /\  E. y ph )  <->  { <. x ,  y >.  |  ( x  e.  A  /\  ph ) }  Fn  A
)
11 eu5 2044 . . . 4  |-  ( E! y ph  <->  ( E. y ph  /\  E* y ph ) )
12 ancom 264 . . . 4  |-  ( ( E. y ph  /\  E* y ph )  <->  ( E* y ph  /\  E. y ph ) )
1311, 12bitri 183 . . 3  |-  ( E! y ph  <->  ( E* y ph  /\  E. y ph ) )
1413ralbii 2439 . 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 5212 . 2  |-  ( F  Fn  A  <->  { <. x ,  y >.  |  ( x  e.  A  /\  ph ) }  Fn  A
)
1710, 14, 163bitr4i 211 1  |-  ( A. x  e.  A  E! y ph  <->  F  Fn  A
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104   A.wal 1329    = wceq 1331   E.wex 1468    e. wcel 1480   E!weu 1997   E*wmo 1998   A.wral 2414   {copab 3983   dom cdm 4534   Fun wfun 5112    Fn wfn 5113
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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-sep 4041  ax-pow 4093  ax-pr 4126
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ral 2419  df-rex 2420  df-v 2683  df-un 3070  df-in 3072  df-ss 3079  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-br 3925  df-opab 3985  df-id 4210  df-xp 4540  df-rel 4541  df-cnv 4542  df-co 4543  df-dm 4544  df-fun 5120  df-fn 5121
This theorem is referenced by:  fnopab  5242  mptfng  5243
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