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Theorem fnoprabg 5945
Description: Functionality and domain of an operation class abstraction. (Contributed by NM, 28-Aug-2007.)
Assertion
Ref Expression
fnoprabg  |-  ( A. x A. y ( ph  ->  E! z ps )  ->  { <. <. x ,  y
>. ,  z >.  |  ( ph  /\  ps ) }  Fn  { <. x ,  y >.  |  ph } )
Distinct variable groups:    x, y,
z    ph, z
Allowed substitution hints:    ph( x, y)    ps( x, y, z)

Proof of Theorem fnoprabg
StepHypRef Expression
1 eumo 2183 . . . . . 6  |-  ( E! z ps  ->  E* z ps )
21imim2i 13 . . . . 5  |-  ( (
ph  ->  E! z ps )  ->  ( ph  ->  E* z ps )
)
3 moanimv 2201 . . . . 5  |-  ( E* z ( ph  /\  ps )  <->  ( ph  ->  E* z ps ) )
42, 3sylibr 203 . . . 4  |-  ( (
ph  ->  E! z ps )  ->  E* z
( ph  /\  ps )
)
542alimi 1547 . . 3  |-  ( A. x A. y ( ph  ->  E! z ps )  ->  A. x A. y E* z ( ph  /\  ps ) )
6 funoprabg 5943 . . 3  |-  ( A. x A. y E* z
( ph  /\  ps )  ->  Fun  { <. <. x ,  y >. ,  z
>.  |  ( ph  /\ 
ps ) } )
75, 6syl 15 . 2  |-  ( A. x A. y ( ph  ->  E! z ps )  ->  Fun  { <. <. x ,  y >. ,  z
>.  |  ( ph  /\ 
ps ) } )
8 dmoprab 5928 . . 3  |-  dom  { <. <. x ,  y
>. ,  z >.  |  ( ph  /\  ps ) }  =  { <. x ,  y >.  |  E. z ( ph  /\ 
ps ) }
9 nfa1 1756 . . . 4  |-  F/ x A. x A. y (
ph  ->  E! z ps )
10 nfa2 1777 . . . 4  |-  F/ y A. x A. y
( ph  ->  E! z ps )
11 simpl 443 . . . . . . . 8  |-  ( (
ph  /\  ps )  ->  ph )
1211exlimiv 1666 . . . . . . 7  |-  ( E. z ( ph  /\  ps )  ->  ph )
13 euex 2166 . . . . . . . . . 10  |-  ( E! z ps  ->  E. z ps )
1413imim2i 13 . . . . . . . . 9  |-  ( (
ph  ->  E! z ps )  ->  ( ph  ->  E. z ps )
)
1514ancld 536 . . . . . . . 8  |-  ( (
ph  ->  E! z ps )  ->  ( ph  ->  ( ph  /\  E. z ps ) ) )
16 19.42v 1846 . . . . . . . 8  |-  ( E. z ( ph  /\  ps )  <->  ( ph  /\  E. z ps ) )
1715, 16syl6ibr 218 . . . . . . 7  |-  ( (
ph  ->  E! z ps )  ->  ( ph  ->  E. z ( ph  /\ 
ps ) ) )
1812, 17impbid2 195 . . . . . 6  |-  ( (
ph  ->  E! z ps )  ->  ( E. z ( ph  /\  ps )  <->  ph ) )
1918sps 1739 . . . . 5  |-  ( A. y ( ph  ->  E! z ps )  -> 
( E. z (
ph  /\  ps )  <->  ph ) )
2019sps 1739 . . . 4  |-  ( A. x A. y ( ph  ->  E! z ps )  ->  ( E. z (
ph  /\  ps )  <->  ph ) )
219, 10, 20opabbid 4081 . . 3  |-  ( A. x A. y ( ph  ->  E! z ps )  ->  { <. x ,  y
>.  |  E. z
( ph  /\  ps ) }  =  { <. x ,  y >.  |  ph } )
228, 21syl5eq 2327 . 2  |-  ( A. x A. y ( ph  ->  E! z ps )  ->  dom  { <. <. x ,  y >. ,  z
>.  |  ( ph  /\ 
ps ) }  =  { <. x ,  y
>.  |  ph } )
23 df-fn 5258 . 2  |-  ( {
<. <. x ,  y
>. ,  z >.  |  ( ph  /\  ps ) }  Fn  { <. x ,  y >.  |  ph } 
<->  ( Fun  { <. <.
x ,  y >. ,  z >.  |  (
ph  /\  ps ) }  /\  dom  { <. <.
x ,  y >. ,  z >.  |  (
ph  /\  ps ) }  =  { <. x ,  y >.  |  ph } ) )
247, 22, 23sylanbrc 645 1  |-  ( A. x A. y ( ph  ->  E! z ps )  ->  { <. <. x ,  y
>. ,  z >.  |  ( ph  /\  ps ) }  Fn  { <. x ,  y >.  |  ph } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358   A.wal 1527   E.wex 1528    = wceq 1623   E!weu 2143   E*wmo 2144   {copab 4076   dom cdm 4689   Fun wfun 5249    Fn wfn 5250   {coprab 5859
This theorem is referenced by:  fnoprab  5947  ovg  5986
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-br 4024  df-opab 4078  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-fun 5257  df-fn 5258  df-oprab 5862
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