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Theorem tfrlemiex 6052
Description: Lemma for tfrlemi1 6053. (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 6050 . . 3  |-  ( ph  ->  B  e.  _V )
7 uniexg 4241 . . 3  |-  ( B  e.  _V  ->  U. B  e.  _V )
86, 7syl 14 . 2  |-  ( ph  ->  U. B  e.  _V )
91, 2, 3, 4, 5tfrlemibfn 6049 . . 3  |-  ( ph  ->  U. B  Fn  x
)
101, 2, 3, 4, 5tfrlemiubacc 6051 . . 3  |-  ( ph  ->  A. u  e.  x  ( U. B `  u
)  =  ( F `
 ( U. B  |`  u ) ) )
119, 10jca 300 . 2  |-  ( ph  ->  ( U. B  Fn  x  /\  A. u  e.  x  ( U. B `  u )  =  ( F `  ( U. B  |`  u ) ) ) )
12 fneq1 5069 . . . 4  |-  ( f  =  U. B  -> 
( f  Fn  x  <->  U. B  Fn  x ) )
13 fveq1 5269 . . . . . 6  |-  ( f  =  U. B  -> 
( f `  u
)  =  ( U. B `  u )
)
14 reseq1 4677 . . . . . . 7  |-  ( f  =  U. B  -> 
( f  |`  u
)  =  ( U. B  |`  u ) )
1514fveq2d 5274 . . . . . 6  |-  ( f  =  U. B  -> 
( F `  (
f  |`  u ) )  =  ( F `  ( U. B  |`  u
) ) )
1613, 15eqeq12d 2099 . . . . 5  |-  ( f  =  U. B  -> 
( ( f `  u )  =  ( F `  ( f  |`  u ) )  <->  ( U. B `  u )  =  ( F `  ( U. B  |`  u
) ) ) )
1716ralbidv 2376 . . . 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 457 . . 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 2700 . 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 61 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 102    /\ w3a 922   A.wal 1285    = 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   Oncon0 4166    |` cres 4415   Fun wfun 4977    Fn wfn 4978   ` cfv 4983
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:  tfrlemi1  6053
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