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Theorem tfr1onlemex 6068
Description: Lemma for tfr1on 6071. (Contributed by Jim Kingdon, 16-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
tfr1onlemex  |-  ( ph  ->  E. f ( f  Fn  D  /\  A. u  e.  D  (
f `  u )  =  ( G `  ( f  |`  u
) ) ) )
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, f, g, h, w, z    u, B, f, w    D, h, w, z, x    u, D    h, G, z, y   
u, G, w    g, X, z    ph, w    y, w
Allowed substitution hints:    ph( y, u)    A( y, w, u)    B( x, y)    D( y)    F( x, y, z, w, u, f, g, h)    G( g)    X( y, w, u, h)

Proof of Theorem tfr1onlemex
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, 9tfr1onlembex 6066 . . 3  |-  ( ph  ->  B  e.  _V )
11 uniexg 4241 . . 3  |-  ( B  e.  _V  ->  U. B  e.  _V )
1210, 11syl 14 . 2  |-  ( ph  ->  U. B  e.  _V )
131, 2, 3, 4, 5, 6, 7, 8, 9tfr1onlembfn 6065 . . 3  |-  ( ph  ->  U. B  Fn  D
)
141, 2, 3, 4, 5, 6, 7, 8, 9tfr1onlemubacc 6067 . . 3  |-  ( ph  ->  A. u  e.  D  ( U. B `  u
)  =  ( G `
 ( U. B  |`  u ) ) )
1513, 14jca 300 . 2  |-  ( ph  ->  ( U. B  Fn  D  /\  A. u  e.  D  ( U. B `  u )  =  ( G `  ( U. B  |`  u ) ) ) )
16 fneq1 5069 . . . 4  |-  ( f  =  U. B  -> 
( f  Fn  D  <->  U. B  Fn  D ) )
17 fveq1 5269 . . . . . 6  |-  ( f  =  U. B  -> 
( f `  u
)  =  ( U. B `  u )
)
18 reseq1 4677 . . . . . . 7  |-  ( f  =  U. B  -> 
( f  |`  u
)  =  ( U. B  |`  u ) )
1918fveq2d 5274 . . . . . 6  |-  ( f  =  U. B  -> 
( G `  (
f  |`  u ) )  =  ( G `  ( U. B  |`  u
) ) )
2017, 19eqeq12d 2099 . . . . 5  |-  ( f  =  U. B  -> 
( ( f `  u )  =  ( G `  ( f  |`  u ) )  <->  ( U. B `  u )  =  ( G `  ( U. B  |`  u
) ) ) )
2120ralbidv 2376 . . . 4  |-  ( f  =  U. B  -> 
( A. u  e.  D  ( f `  u )  =  ( G `  ( f  |`  u ) )  <->  A. u  e.  D  ( U. B `  u )  =  ( G `  ( U. B  |`  u
) ) ) )
2216, 21anbi12d 457 . . 3  |-  ( f  =  U. B  -> 
( ( f  Fn  D  /\  A. u  e.  D  ( f `  u )  =  ( G `  ( f  |`  u ) ) )  <-> 
( U. B  Fn  D  /\  A. u  e.  D  ( U. B `  u )  =  ( G `  ( U. B  |`  u ) ) ) ) )
2322spcegv 2700 . 2  |-  ( U. B  e.  _V  ->  ( ( U. B  Fn  D  /\  A. u  e.  D  ( U. B `  u )  =  ( G `  ( U. B  |`  u ) ) )  ->  E. f
( f  Fn  D  /\  A. u  e.  D  ( f `  u
)  =  ( G `
 ( f  |`  u ) ) ) ) )
2412, 15, 23sylc 61 1  |-  ( ph  ->  E. f ( f  Fn  D  /\  A. u  e.  D  (
f `  u )  =  ( G `  ( f  |`  u
) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    /\ w3a 922    = wceq 1287   E.wex 1424    e. wcel 1436   {cab 2071   A.wral 2355   E.wrex 2356   _Vcvv 2615    u. cun 2986   {csn 3431   <.cop 3434   U.cuni 3638   Ord word 4165   suc csuc 4168    |` cres 4415   Fun wfun 4977    Fn wfn 4978   ` cfv 4983  recscrecs 6025
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 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-13 1447  ax-14 1448  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067  ax-coll 3931  ax-sep 3934  ax-pow 3986  ax-pr 4012  ax-un 4236  ax-setind 4328
This theorem depends on definitions:  df-bi 115  df-3an 924  df-tru 1290  df-fal 1293  df-nf 1393  df-sb 1690  df-eu 1948  df-mo 1949  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-ne 2252  df-ral 2360  df-rex 2361  df-reu 2362  df-rab 2364  df-v 2617  df-sbc 2830  df-csb 2923  df-dif 2990  df-un 2992  df-in 2994  df-ss 3001  df-nul 3276  df-pw 3417  df-sn 3437  df-pr 3438  df-op 3440  df-uni 3639  df-iun 3717  df-br 3823  df-opab 3877  df-mpt 3878  df-tr 3914  df-id 4096  df-iord 4169  df-on 4171  df-suc 4174  df-xp 4419  df-rel 4420  df-cnv 4421  df-co 4422  df-dm 4423  df-rn 4424  df-res 4425  df-ima 4426  df-iota 4948  df-fun 4985  df-fn 4986  df-f 4987  df-f1 4988  df-fo 4989  df-f1o 4990  df-fv 4991  df-recs 6026
This theorem is referenced by:  tfr1onlemaccex  6069
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