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Theorem tfrlemisucfn 5993
Description: We can extend an acceptable function by one element to produce a function. Lemma for tfrlemi1 6001. (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 2613 . . 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 5981 . . 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 2083 . 2  |-  ( g  u.  { <. z ,  ( F `  g ) >. } )  =  ( g  u. 
{ <. z ,  ( F `  g )
>. } )
8 df-suc 4154 . 2  |-  suc  z  =  ( z  u. 
{ z } )
9 elirrv 4319 . . 3  |-  -.  z  e.  z
109a1i 9 . 2  |-  ( ph  ->  -.  z  e.  z )
112, 5, 6, 7, 8, 10fnunsn 5057 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 1283    = wceq 1285    e. wcel 1434   {cab 2069   A.wral 2353   E.wrex 2354   _Vcvv 2610    u. cun 2980   {csn 3416   <.cop 3419   Oncon0 4146   suc csuc 4148    |` cres 4393   Fun wfun 4946    Fn wfn 4947   ` cfv 4952
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 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-sep 3916  ax-pow 3968  ax-pr 3992  ax-setind 4308
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ne 2250  df-ral 2358  df-rex 2359  df-v 2612  df-dif 2984  df-un 2986  df-in 2988  df-ss 2995  df-nul 3268  df-pw 3402  df-sn 3422  df-pr 3423  df-op 3425  df-uni 3622  df-br 3806  df-opab 3860  df-id 4076  df-suc 4154  df-xp 4397  df-rel 4398  df-cnv 4399  df-co 4400  df-dm 4401  df-iota 4917  df-fun 4954  df-fn 4955  df-fv 4960
This theorem is referenced by:  tfrlemisucaccv  5994  tfrlemibfn  5997
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