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Theorem tfr1onlembex 6102
Description: Lemma for tfr1on 6107. The set  B exists. (Contributed by Jim Kingdon, 14-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
) ) ) }
tfr1onlembacc.3  |-  B  =  { h  |  E. z  e.  D  E. g ( g  Fn  z  /\  g  e.  A  /\  h  =  ( g  u.  { <. z ,  ( G `
 g ) >. } ) ) }
tfr1onlembacc.u  |-  ( (
ph  /\  x  e.  U. X )  ->  suc  x  e.  X )
tfr1onlembacc.4  |-  ( ph  ->  D  e.  X )
tfr1onlembacc.5  |-  ( ph  ->  A. z  e.  D  E. g ( g  Fn  z  /\  A. w  e.  z  ( g `  w )  =  ( G `  ( g  |`  w ) ) ) )
Assertion
Ref Expression
tfr1onlembex  |-  ( ph  ->  B  e.  _V )
Distinct variable groups:    A, f, g, h, x, z    D, f, g, x    f, G, x, y    f, X, x    ph, f, g, h, x, z    y, g, z    B, g, h, w, z    D, h, z    h, G, w, z, f, y, x    g, X, z
Allowed substitution hints:    ph( y, w)    A( y, w)    B( x, y, f)    D( y, w)    F( x, y, z, w, f, g, h)    G( g)    X( y, w, h)

Proof of Theorem tfr1onlembex
StepHypRef Expression
1 tfr1on.f . . . 4  |-  F  = recs ( G )
2 tfr1on.g . . . 4  |-  ( ph  ->  Fun  G )
3 tfr1on.x . . . 4  |-  ( ph  ->  Ord  X )
4 tfr1on.ex . . . 4  |-  ( (
ph  /\  x  e.  X  /\  f  Fn  x
)  ->  ( G `  f )  e.  _V )
5 tfr1onlemsucfn.1 . . . 4  |-  A  =  { f  |  E. x  e.  X  (
f  Fn  x  /\  A. y  e.  x  ( f `  y )  =  ( G `  ( f  |`  y
) ) ) }
6 tfr1onlembacc.3 . . . 4  |-  B  =  { h  |  E. z  e.  D  E. g ( g  Fn  z  /\  g  e.  A  /\  h  =  ( g  u.  { <. z ,  ( G `
 g ) >. } ) ) }
7 tfr1onlembacc.u . . . 4  |-  ( (
ph  /\  x  e.  U. X )  ->  suc  x  e.  X )
8 tfr1onlembacc.4 . . . 4  |-  ( ph  ->  D  e.  X )
9 tfr1onlembacc.5 . . . 4  |-  ( ph  ->  A. z  e.  D  E. g ( g  Fn  z  /\  A. w  e.  z  ( g `  w )  =  ( G `  ( g  |`  w ) ) ) )
101, 2, 3, 4, 5, 6, 7, 8, 9tfr1onlembfn 6101 . . 3  |-  ( ph  ->  U. B  Fn  D
)
11 fnex 5511 . . 3  |-  ( ( U. B  Fn  D  /\  D  e.  X
)  ->  U. B  e. 
_V )
1210, 8, 11syl2anc 403 . 2  |-  ( ph  ->  U. B  e.  _V )
13 uniexb 4293 . 2  |-  ( B  e.  _V  <->  U. B  e. 
_V )
1412, 13sylibr 132 1  |-  ( ph  ->  B  e.  _V )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    /\ w3a 924    = wceq 1289   E.wex 1426    e. wcel 1438   {cab 2074   A.wral 2359   E.wrex 2360   _Vcvv 2619    u. cun 2997   {csn 3444   <.cop 3447   U.cuni 3651   Ord word 4187   suc csuc 4190    |` cres 4438   Fun wfun 5004    Fn wfn 5005   ` cfv 5010  recscrecs 6061
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-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-coll 3952  ax-sep 3955  ax-pow 4007  ax-pr 4034  ax-un 4258  ax-setind 4351
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-reu 2366  df-rab 2368  df-v 2621  df-sbc 2841  df-csb 2934  df-dif 3001  df-un 3003  df-in 3005  df-ss 3012  df-nul 3287  df-pw 3429  df-sn 3450  df-pr 3451  df-op 3453  df-uni 3652  df-iun 3730  df-br 3844  df-opab 3898  df-mpt 3899  df-tr 3935  df-id 4118  df-iord 4191  df-on 4193  df-suc 4196  df-xp 4442  df-rel 4443  df-cnv 4444  df-co 4445  df-dm 4446  df-rn 4447  df-res 4448  df-ima 4449  df-iota 4975  df-fun 5012  df-fn 5013  df-f 5014  df-f1 5015  df-fo 5016  df-f1o 5017  df-fv 5018  df-recs 6062
This theorem is referenced by:  tfr1onlemex  6104
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