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Theorem tfrlemiex 6180
Description: Lemma for tfrlemi1 6181. (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 6178 . . 3  |-  ( ph  ->  B  e.  _V )
7 uniexg 4319 . . 3  |-  ( B  e.  _V  ->  U. B  e.  _V )
86, 7syl 14 . 2  |-  ( ph  ->  U. B  e.  _V )
91, 2, 3, 4, 5tfrlemibfn 6177 . . 3  |-  ( ph  ->  U. B  Fn  x
)
101, 2, 3, 4, 5tfrlemiubacc 6179 . . 3  |-  ( ph  ->  A. u  e.  x  ( U. B `  u
)  =  ( F `
 ( U. B  |`  u ) ) )
119, 10jca 302 . 2  |-  ( ph  ->  ( U. B  Fn  x  /\  A. u  e.  x  ( U. B `  u )  =  ( F `  ( U. B  |`  u ) ) ) )
12 fneq1 5167 . . . 4  |-  ( f  =  U. B  -> 
( f  Fn  x  <->  U. B  Fn  x ) )
13 fveq1 5372 . . . . . 6  |-  ( f  =  U. B  -> 
( f `  u
)  =  ( U. B `  u )
)
14 reseq1 4769 . . . . . . 7  |-  ( f  =  U. B  -> 
( f  |`  u
)  =  ( U. B  |`  u ) )
1514fveq2d 5377 . . . . . 6  |-  ( f  =  U. B  -> 
( F `  (
f  |`  u ) )  =  ( F `  ( U. B  |`  u
) ) )
1613, 15eqeq12d 2127 . . . . 5  |-  ( f  =  U. B  -> 
( ( f `  u )  =  ( F `  ( f  |`  u ) )  <->  ( U. B `  u )  =  ( F `  ( U. B  |`  u
) ) ) )
1716ralbidv 2409 . . . 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 462 . . 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 2743 . 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 103    /\ w3a 943   A.wal 1310    = wceq 1312   E.wex 1449    e. wcel 1461   {cab 2099   A.wral 2388   E.wrex 2389   _Vcvv 2655    u. cun 3033   {csn 3491   <.cop 3494   U.cuni 3700   Oncon0 4243    |` cres 4499   Fun wfun 5073    Fn wfn 5074   ` cfv 5079
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 586  ax-in2 587  ax-io 681  ax-5 1404  ax-7 1405  ax-gen 1406  ax-ie1 1450  ax-ie2 1451  ax-8 1463  ax-10 1464  ax-11 1465  ax-i12 1466  ax-bndl 1467  ax-4 1468  ax-13 1472  ax-14 1473  ax-17 1487  ax-i9 1491  ax-ial 1495  ax-i5r 1496  ax-ext 2095  ax-coll 4001  ax-sep 4004  ax-pow 4056  ax-pr 4089  ax-un 4313  ax-setind 4410
This theorem depends on definitions:  df-bi 116  df-3an 945  df-tru 1315  df-fal 1318  df-nf 1418  df-sb 1717  df-eu 1976  df-mo 1977  df-clab 2100  df-cleq 2106  df-clel 2109  df-nfc 2242  df-ne 2281  df-ral 2393  df-rex 2394  df-reu 2395  df-rab 2397  df-v 2657  df-sbc 2877  df-csb 2970  df-dif 3037  df-un 3039  df-in 3041  df-ss 3048  df-nul 3328  df-pw 3476  df-sn 3497  df-pr 3498  df-op 3500  df-uni 3701  df-iun 3779  df-br 3894  df-opab 3948  df-mpt 3949  df-tr 3985  df-id 4173  df-iord 4246  df-on 4248  df-suc 4251  df-xp 4503  df-rel 4504  df-cnv 4505  df-co 4506  df-dm 4507  df-rn 4508  df-res 4509  df-ima 4510  df-iota 5044  df-fun 5081  df-fn 5082  df-f 5083  df-f1 5084  df-fo 5085  df-f1o 5086  df-fv 5087  df-recs 6154
This theorem is referenced by:  tfrlemi1  6181
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