ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  tfr1onlemsucfn Unicode version

Theorem tfr1onlemsucfn 6037
Description: We can extend an acceptable function by one element to produce a function. Lemma for tfr1on 6047. (Contributed by Jim Kingdon, 12-Mar-2022.)
Hypotheses
Ref Expression
tfr1on.f  |-  F  = recs ( G )
tfr1on.g  |-  ( ph  ->  Fun  G )
tfr1on.x  |-  ( ph  ->  Ord  X )
tfr1on.ex  |-  ( (
ph  /\  x  e.  X  /\  f  Fn  x
)  ->  ( G `  f )  e.  _V )
tfr1onlemsucfn.1  |-  A  =  { f  |  E. x  e.  X  (
f  Fn  x  /\  A. y  e.  x  ( f `  y )  =  ( G `  ( f  |`  y
) ) ) }
tfr1onlemsucfn.3  |-  ( ph  ->  z  e.  X )
tfr1onlemsucfn.4  |-  ( ph  ->  g  Fn  z )
tfr1onlemsucfn.5  |-  ( ph  ->  g  e.  A )
Assertion
Ref Expression
tfr1onlemsucfn  |-  ( ph  ->  ( g  u.  { <. z ,  ( G `
 g ) >. } )  Fn  suc  z )
Distinct variable groups:    f, G, x   
f, X, x    f,
g    ph, f, x    z,
f, x
Allowed substitution hints:    ph( y, z, g)    A( x, y, z, f, g)    F( x, y, z, f, g)    G( y, z, g)    X( y, z, g)

Proof of Theorem tfr1onlemsucfn
StepHypRef Expression
1 tfr1onlemsucfn.3 . . 3  |-  ( ph  ->  z  e.  X )
21elexd 2623 . 2  |-  ( ph  ->  z  e.  _V )
3 fneq2 5056 . . . . . 6  |-  ( x  =  z  ->  (
f  Fn  x  <->  f  Fn  z ) )
43imbi1d 229 . . . . 5  |-  ( x  =  z  ->  (
( f  Fn  x  ->  ( G `  f
)  e.  _V )  <->  ( f  Fn  z  -> 
( G `  f
)  e.  _V )
) )
54albidv 1747 . . . 4  |-  ( x  =  z  ->  ( A. f ( f  Fn  x  ->  ( G `  f )  e.  _V ) 
<-> 
A. f ( f  Fn  z  ->  ( G `  f )  e.  _V ) ) )
6 tfr1on.ex . . . . . . 7  |-  ( (
ph  /\  x  e.  X  /\  f  Fn  x
)  ->  ( G `  f )  e.  _V )
763expia 1141 . . . . . 6  |-  ( (
ph  /\  x  e.  X )  ->  (
f  Fn  x  -> 
( G `  f
)  e.  _V )
)
87alrimiv 1797 . . . . 5  |-  ( (
ph  /\  x  e.  X )  ->  A. f
( f  Fn  x  ->  ( G `  f
)  e.  _V )
)
98ralrimiva 2440 . . . 4  |-  ( ph  ->  A. x  e.  X  A. f ( f  Fn  x  ->  ( G `  f )  e.  _V ) )
105, 9, 1rspcdva 2717 . . 3  |-  ( ph  ->  A. f ( f  Fn  z  ->  ( G `  f )  e.  _V ) )
11 tfr1onlemsucfn.4 . . 3  |-  ( ph  ->  g  Fn  z )
12 fneq1 5055 . . . . 5  |-  ( f  =  g  ->  (
f  Fn  z  <->  g  Fn  z ) )
13 fveq2 5253 . . . . . 6  |-  ( f  =  g  ->  ( G `  f )  =  ( G `  g ) )
1413eleq1d 2151 . . . . 5  |-  ( f  =  g  ->  (
( G `  f
)  e.  _V  <->  ( G `  g )  e.  _V ) )
1512, 14imbi12d 232 . . . 4  |-  ( f  =  g  ->  (
( f  Fn  z  ->  ( G `  f
)  e.  _V )  <->  ( g  Fn  z  -> 
( G `  g
)  e.  _V )
) )
1615spv 1783 . . 3  |-  ( A. f ( f  Fn  z  ->  ( G `  f )  e.  _V )  ->  ( g  Fn  z  ->  ( G `  g )  e.  _V ) )
1710, 11, 16sylc 61 . 2  |-  ( ph  ->  ( G `  g
)  e.  _V )
18 eqid 2083 . 2  |-  ( g  u.  { <. z ,  ( G `  g ) >. } )  =  ( g  u. 
{ <. z ,  ( G `  g )
>. } )
19 df-suc 4162 . 2  |-  suc  z  =  ( z  u. 
{ z } )
20 tfr1on.x . . . 4  |-  ( ph  ->  Ord  X )
21 ordelon 4174 . . . 4  |-  ( ( Ord  X  /\  z  e.  X )  ->  z  e.  On )
2220, 1, 21syl2anc 403 . . 3  |-  ( ph  ->  z  e.  On )
23 eloni 4166 . . 3  |-  ( z  e.  On  ->  Ord  z )
24 ordirr 4321 . . 3  |-  ( Ord  z  ->  -.  z  e.  z )
2522, 23, 243syl 17 . 2  |-  ( ph  ->  -.  z  e.  z )
262, 17, 11, 18, 19, 25fnunsn 5074 1  |-  ( ph  ->  ( g  u.  { <. z ,  ( G `
 g ) >. } )  Fn  suc  z )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 102    /\ w3a 920   A.wal 1283    = wceq 1285    e. wcel 1434   {cab 2069   A.wral 2353   E.wrex 2354   _Vcvv 2612    u. cun 2982   {csn 3422   <.cop 3425   Ord word 4153   Oncon0 4154   suc csuc 4156    |` cres 4403   Fun wfun 4963    Fn wfn 4964   ` cfv 4969  recscrecs 6001
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 3922  ax-pow 3974  ax-pr 4000  ax-setind 4316
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 2614  df-dif 2986  df-un 2988  df-in 2990  df-ss 2997  df-nul 3270  df-pw 3408  df-sn 3428  df-pr 3429  df-op 3431  df-uni 3628  df-br 3812  df-opab 3866  df-tr 3902  df-id 4084  df-iord 4157  df-on 4159  df-suc 4162  df-xp 4407  df-rel 4408  df-cnv 4409  df-co 4410  df-dm 4411  df-iota 4934  df-fun 4971  df-fn 4972  df-fv 4977
This theorem is referenced by:  tfr1onlemsucaccv  6038  tfr1onlembfn  6041
  Copyright terms: Public domain W3C validator