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Theorem tfrlemiex 6322
Description: Lemma for tfrlemi1 6323. (Contributed by Jim Kingdon, 18-Mar-2019.) (Proof shortened by Mario Carneiro, 24-May-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 ) )
tfrlemi1.3  |-  B  =  { h  |  E. z  e.  x  E. g ( g  Fn  z  /\  g  e.  A  /\  h  =  ( g  u.  { <. z ,  ( F `
 g ) >. } ) ) }
tfrlemi1.4  |-  ( ph  ->  x  e.  On )
tfrlemi1.5  |-  ( ph  ->  A. z  e.  x  E. g ( g  Fn  z  /\  A. w  e.  z  ( g `  w )  =  ( F `  ( g  |`  w ) ) ) )
Assertion
Ref Expression
tfrlemiex  |-  ( ph  ->  E. f ( f  Fn  x  /\  A. u  e.  x  (
f `  u )  =  ( F `  ( f  |`  u
) ) ) )
Distinct variable groups:    f, g, h, u, w, x, y, z, A    f, F, g, h, u, w, x, y, z    ph, w, y    u, B, w, f, g, h, z    ph, g, h, z
Allowed substitution hints:    ph( x, u, f)    B( x, y)

Proof of Theorem tfrlemiex
StepHypRef Expression
1 tfrlemisucfn.1 . . . 4  |-  A  =  { f  |  E. x  e.  On  (
f  Fn  x  /\  A. y  e.  x  ( f `  y )  =  ( F `  ( f  |`  y
) ) ) }
2 tfrlemisucfn.2 . . . 4  |-  ( ph  ->  A. x ( Fun 
F  /\  ( F `  x )  e.  _V ) )
3 tfrlemi1.3 . . . 4  |-  B  =  { h  |  E. z  e.  x  E. g ( g  Fn  z  /\  g  e.  A  /\  h  =  ( g  u.  { <. z ,  ( F `
 g ) >. } ) ) }
4 tfrlemi1.4 . . . 4  |-  ( ph  ->  x  e.  On )
5 tfrlemi1.5 . . . 4  |-  ( ph  ->  A. z  e.  x  E. g ( g  Fn  z  /\  A. w  e.  z  ( g `  w )  =  ( F `  ( g  |`  w ) ) ) )
61, 2, 3, 4, 5tfrlemibex 6320 . . 3  |-  ( ph  ->  B  e.  _V )
7 uniexg 4433 . . 3  |-  ( B  e.  _V  ->  U. B  e.  _V )
86, 7syl 14 . 2  |-  ( ph  ->  U. B  e.  _V )
91, 2, 3, 4, 5tfrlemibfn 6319 . . 3  |-  ( ph  ->  U. B  Fn  x
)
101, 2, 3, 4, 5tfrlemiubacc 6321 . . 3  |-  ( ph  ->  A. u  e.  x  ( U. B `  u
)  =  ( F `
 ( U. B  |`  u ) ) )
119, 10jca 306 . 2  |-  ( ph  ->  ( U. B  Fn  x  /\  A. u  e.  x  ( U. B `  u )  =  ( F `  ( U. B  |`  u ) ) ) )
12 fneq1 5296 . . . 4  |-  ( f  =  U. B  -> 
( f  Fn  x  <->  U. B  Fn  x ) )
13 fveq1 5506 . . . . . 6  |-  ( f  =  U. B  -> 
( f `  u
)  =  ( U. B `  u )
)
14 reseq1 4894 . . . . . . 7  |-  ( f  =  U. B  -> 
( f  |`  u
)  =  ( U. B  |`  u ) )
1514fveq2d 5511 . . . . . 6  |-  ( f  =  U. B  -> 
( F `  (
f  |`  u ) )  =  ( F `  ( U. B  |`  u
) ) )
1613, 15eqeq12d 2190 . . . . 5  |-  ( f  =  U. B  -> 
( ( f `  u )  =  ( F `  ( f  |`  u ) )  <->  ( U. B `  u )  =  ( F `  ( U. B  |`  u
) ) ) )
1716ralbidv 2475 . . . 4  |-  ( f  =  U. B  -> 
( A. u  e.  x  ( f `  u )  =  ( F `  ( f  |`  u ) )  <->  A. u  e.  x  ( U. B `  u )  =  ( F `  ( U. B  |`  u
) ) ) )
1812, 17anbi12d 473 . . 3  |-  ( f  =  U. B  -> 
( ( f  Fn  x  /\  A. u  e.  x  ( f `  u )  =  ( F `  ( f  |`  u ) ) )  <-> 
( U. B  Fn  x  /\  A. u  e.  x  ( U. B `  u )  =  ( F `  ( U. B  |`  u ) ) ) ) )
1918spcegv 2823 . 2  |-  ( U. B  e.  _V  ->  ( ( U. B  Fn  x  /\  A. u  e.  x  ( U. B `  u )  =  ( F `  ( U. B  |`  u ) ) )  ->  E. f
( f  Fn  x  /\  A. u  e.  x  ( f `  u
)  =  ( F `
 ( f  |`  u ) ) ) ) )
208, 11, 19sylc 62 1  |-  ( ph  ->  E. f ( f  Fn  x  /\  A. u  e.  x  (
f `  u )  =  ( F `  ( f  |`  u
) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    /\ w3a 978   A.wal 1351    = wceq 1353   E.wex 1490    e. wcel 2146   {cab 2161   A.wral 2453   E.wrex 2454   _Vcvv 2735    u. cun 3125   {csn 3589   <.cop 3592   U.cuni 3805   Oncon0 4357    |` cres 4622   Fun wfun 5202    Fn wfn 5203   ` cfv 5208
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1445  ax-7 1446  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-10 1503  ax-11 1504  ax-i12 1505  ax-bndl 1507  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532  ax-i5r 1533  ax-13 2148  ax-14 2149  ax-ext 2157  ax-coll 4113  ax-sep 4116  ax-pow 4169  ax-pr 4203  ax-un 4427  ax-setind 4530
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1459  df-sb 1761  df-eu 2027  df-mo 2028  df-clab 2162  df-cleq 2168  df-clel 2171  df-nfc 2306  df-ne 2346  df-ral 2458  df-rex 2459  df-reu 2460  df-rab 2462  df-v 2737  df-sbc 2961  df-csb 3056  df-dif 3129  df-un 3131  df-in 3133  df-ss 3140  df-nul 3421  df-pw 3574  df-sn 3595  df-pr 3596  df-op 3598  df-uni 3806  df-iun 3884  df-br 3999  df-opab 4060  df-mpt 4061  df-tr 4097  df-id 4287  df-iord 4360  df-on 4362  df-suc 4365  df-xp 4626  df-rel 4627  df-cnv 4628  df-co 4629  df-dm 4630  df-rn 4631  df-res 4632  df-ima 4633  df-iota 5170  df-fun 5210  df-fn 5211  df-f 5212  df-f1 5213  df-fo 5214  df-f1o 5215  df-fv 5216  df-recs 6296
This theorem is referenced by:  tfrlemi1  6323
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