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Theorem tfrlemisucfn 6089
Description: We can extend an acceptable function by one element to produce a function. Lemma for tfrlemi1 6097. (Contributed by Jim Kingdon, 2-Jul-2019.)
Hypotheses
Ref Expression
tfrlemisucfn.1  |-  A  =  { f  |  E. x  e.  On  (
f  Fn  x  /\  A. y  e.  x  ( f `  y )  =  ( F `  ( f  |`  y
) ) ) }
tfrlemisucfn.2  |-  ( ph  ->  A. x ( Fun 
F  /\  ( F `  x )  e.  _V ) )
tfrlemisucfn.3  |-  ( ph  ->  z  e.  On )
tfrlemisucfn.4  |-  ( ph  ->  g  Fn  z )
tfrlemisucfn.5  |-  ( ph  ->  g  e.  A )
Assertion
Ref Expression
tfrlemisucfn  |-  ( ph  ->  ( g  u.  { <. z ,  ( F `
 g ) >. } )  Fn  suc  z )
Distinct variable groups:    f, g, x, y, z, A    f, F, g, x, y, z    ph, y
Allowed substitution hints:    ph( x, z, f, g)

Proof of Theorem tfrlemisucfn
StepHypRef Expression
1 vex 2622 . . 3  |-  z  e. 
_V
21a1i 9 . 2  |-  ( ph  ->  z  e.  _V )
3 tfrlemisucfn.2 . . . 4  |-  ( ph  ->  A. x ( Fun 
F  /\  ( F `  x )  e.  _V ) )
43tfrlem3-2d 6077 . . 3  |-  ( ph  ->  ( Fun  F  /\  ( F `  g )  e.  _V ) )
54simprd 112 . 2  |-  ( ph  ->  ( F `  g
)  e.  _V )
6 tfrlemisucfn.4 . 2  |-  ( ph  ->  g  Fn  z )
7 eqid 2088 . 2  |-  ( g  u.  { <. z ,  ( F `  g ) >. } )  =  ( g  u. 
{ <. z ,  ( F `  g )
>. } )
8 df-suc 4198 . 2  |-  suc  z  =  ( z  u. 
{ z } )
9 elirrv 4364 . . 3  |-  -.  z  e.  z
109a1i 9 . 2  |-  ( ph  ->  -.  z  e.  z )
112, 5, 6, 7, 8, 10fnunsn 5121 1  |-  ( ph  ->  ( g  u.  { <. z ,  ( F `
 g ) >. } )  Fn  suc  z )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 102   A.wal 1287    = wceq 1289    e. wcel 1438   {cab 2074   A.wral 2359   E.wrex 2360   _Vcvv 2619    u. cun 2997   {csn 3446   <.cop 3449   Oncon0 4190   suc csuc 4192    |` cres 4440   Fun wfun 5009    Fn wfn 5010   ` cfv 5015
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-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3957  ax-pow 4009  ax-pr 4036  ax-setind 4353
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ne 2256  df-ral 2364  df-rex 2365  df-v 2621  df-dif 3001  df-un 3003  df-in 3005  df-ss 3012  df-nul 3287  df-pw 3431  df-sn 3452  df-pr 3453  df-op 3455  df-uni 3654  df-br 3846  df-opab 3900  df-id 4120  df-suc 4198  df-xp 4444  df-rel 4445  df-cnv 4446  df-co 4447  df-dm 4448  df-iota 4980  df-fun 5017  df-fn 5018  df-fv 5023
This theorem is referenced by:  tfrlemisucaccv  6090  tfrlemibfn  6093
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