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Theorem tfr1onlembex 6235
Description: Lemma for tfr1on 6240. 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 6234 . . 3  |-  ( ph  ->  U. B  Fn  D
)
11 fnex 5635 . . 3  |-  ( ( U. B  Fn  D  /\  D  e.  X
)  ->  U. B  e. 
_V )
1210, 8, 11syl2anc 408 . 2  |-  ( ph  ->  U. B  e.  _V )
13 uniexb 4389 . 2  |-  ( B  e.  _V  <->  U. B  e. 
_V )
1412, 13sylibr 133 1  |-  ( ph  ->  B  e.  _V )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    /\ w3a 962    = wceq 1331   E.wex 1468    e. wcel 1480   {cab 2123   A.wral 2414   E.wrex 2415   _Vcvv 2681    u. cun 3064   {csn 3522   <.cop 3525   U.cuni 3731   Ord word 4279   suc csuc 4282    |` cres 4536   Fun wfun 5112    Fn wfn 5113   ` cfv 5118  recscrecs 6194
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-coll 4038  ax-sep 4041  ax-pow 4093  ax-pr 4126  ax-un 4350  ax-setind 4447
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ne 2307  df-ral 2419  df-rex 2420  df-reu 2421  df-rab 2423  df-v 2683  df-sbc 2905  df-csb 2999  df-dif 3068  df-un 3070  df-in 3072  df-ss 3079  df-nul 3359  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-uni 3732  df-iun 3810  df-br 3925  df-opab 3985  df-mpt 3986  df-tr 4022  df-id 4210  df-iord 4283  df-on 4285  df-suc 4288  df-xp 4540  df-rel 4541  df-cnv 4542  df-co 4543  df-dm 4544  df-rn 4545  df-res 4546  df-ima 4547  df-iota 5083  df-fun 5120  df-fn 5121  df-f 5122  df-f1 5123  df-fo 5124  df-f1o 5125  df-fv 5126  df-recs 6195
This theorem is referenced by:  tfr1onlemex  6237
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