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Theorem tfrlemisucfn 6377
Description: We can extend an acceptable function by one element to produce a function. Lemma for tfrlemi1 6385. (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 2763 . . 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 6365 . . 3  |-  ( ph  ->  ( Fun  F  /\  ( F `  g )  e.  _V ) )
54simprd 114 . 2  |-  ( ph  ->  ( F `  g
)  e.  _V )
6 tfrlemisucfn.4 . 2  |-  ( ph  ->  g  Fn  z )
7 eqid 2193 . 2  |-  ( g  u.  { <. z ,  ( F `  g ) >. } )  =  ( g  u. 
{ <. z ,  ( F `  g )
>. } )
8 df-suc 4402 . 2  |-  suc  z  =  ( z  u. 
{ z } )
9 elirrv 4580 . . 3  |-  -.  z  e.  z
109a1i 9 . 2  |-  ( ph  ->  -.  z  e.  z )
112, 5, 6, 7, 8, 10fnunsn 5361 1  |-  ( ph  ->  ( g  u.  { <. z ,  ( F `
 g ) >. } )  Fn  suc  z )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 104   A.wal 1362    = wceq 1364    e. wcel 2164   {cab 2179   A.wral 2472   E.wrex 2473   _Vcvv 2760    u. cun 3151   {csn 3618   <.cop 3621   Oncon0 4394   suc csuc 4396    |` cres 4661   Fun wfun 5248    Fn wfn 5249   ` cfv 5254
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2167  ax-ext 2175  ax-sep 4147  ax-pow 4203  ax-pr 4238  ax-setind 4569
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-ral 2477  df-rex 2478  df-v 2762  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3447  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-br 4030  df-opab 4091  df-id 4324  df-suc 4402  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-iota 5215  df-fun 5256  df-fn 5257  df-fv 5262
This theorem is referenced by:  tfrlemisucaccv  6378  tfrlemibfn  6381
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