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Theorem fnopabg 5711
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 2380 . . . . . 6  |-  ( E* y ( x  e.  A  /\  ph )  <->  ( x  e.  A  ->  E* y ph ) )
21albii 1699 . . . . 5  |-  ( A. x E* y ( x  e.  A  /\  ph ) 
<-> 
A. x ( x  e.  A  ->  E* y ph ) )
3 funopab 5622 . . . . 5  |-  ( Fun 
{ <. x ,  y
>.  |  ( x  e.  A  /\  ph ) } 
<-> 
A. x E* y
( x  e.  A  /\  ph ) )
4 df-ral 2761 . . . . 5  |-  ( A. x  e.  A  E* y ph  <->  A. x ( x  e.  A  ->  E* y ph ) )
52, 3, 43bitr4ri 286 . . . 4  |-  ( A. x  e.  A  E* y ph  <->  Fun  { <. x ,  y >.  |  ( x  e.  A  /\  ph ) } )
6 dmopab3 5053 . . . 4  |-  ( A. x  e.  A  E. y ph  <->  dom  { <. x ,  y >.  |  ( x  e.  A  /\  ph ) }  =  A )
75, 6anbi12i 711 . . 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 2904 . . 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 5592 . . 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 285 . 2  |-  ( A. x  e.  A  ( E* y ph  /\  E. y ph )  <->  { <. x ,  y >.  |  ( x  e.  A  /\  ph ) }  Fn  A
)
11 eu5 2345 . . . 4  |-  ( E! y ph  <->  ( E. y ph  /\  E* y ph ) )
12 ancom 457 . . . 4  |-  ( ( E. y ph  /\  E* y ph )  <->  ( E* y ph  /\  E. y ph ) )
1311, 12bitri 257 . . 3  |-  ( E! y ph  <->  ( E* y ph  /\  E. y ph ) )
1413ralbii 2823 . 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 5680 . 2  |-  ( F  Fn  A  <->  { <. x ,  y >.  |  ( x  e.  A  /\  ph ) }  Fn  A
)
1710, 14, 163bitr4i 285 1  |-  ( A. x  e.  A  E! y ph  <->  F  Fn  A
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 189    /\ wa 376   A.wal 1450    = wceq 1452   E.wex 1671    e. wcel 1904   E!weu 2319   E*wmo 2320   A.wral 2756   {copab 4453   dom cdm 4839   Fun wfun 5583    Fn wfn 5584
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518  ax-nul 4527  ax-pr 4639
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-rab 2765  df-v 3033  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-if 3873  df-sn 3960  df-pr 3962  df-op 3966  df-br 4396  df-opab 4455  df-id 4754  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-fun 5591  df-fn 5592
This theorem is referenced by:  fnopab  5712  mptfng  5713  axcontlem2  25074
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