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Theorem tfrlemibex 6192
Description: The set  B exists. Lemma for tfrlemi1 6195. (Contributed by Jim Kingdon, 17-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
tfrlemibex  |-  ( ph  ->  B  e.  _V )
Distinct variable groups:    f, g, h, w, x, y, z, A    f, F, g, h, w, x, y, z    ph, w, y    w, B, f, g, h, z    ph, g, h, z
Allowed substitution hints:    ph( x, f)    B( x, y)

Proof of Theorem tfrlemibex
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, 5tfrlemibfn 6191 . . 3  |-  ( ph  ->  U. B  Fn  x
)
7 vex 2661 . . 3  |-  x  e. 
_V
8 fnex 5608 . . 3  |-  ( ( U. B  Fn  x  /\  x  e.  _V )  ->  U. B  e.  _V )
96, 7, 8sylancl 407 . 2  |-  ( ph  ->  U. B  e.  _V )
10 uniexb 4362 . 2  |-  ( B  e.  _V  <->  U. B  e. 
_V )
119, 10sylibr 133 1  |-  ( ph  ->  B  e.  _V )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    /\ w3a 945   A.wal 1312    = wceq 1314   E.wex 1451    e. wcel 1463   {cab 2101   A.wral 2391   E.wrex 2392   _Vcvv 2658    u. cun 3037   {csn 3495   <.cop 3498   U.cuni 3704   Oncon0 4253    |` cres 4509   Fun wfun 5085    Fn wfn 5086   ` cfv 5091
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 586  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-13 1474  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-coll 4011  ax-sep 4014  ax-pow 4066  ax-pr 4099  ax-un 4323  ax-setind 4420
This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-fal 1320  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ne 2284  df-ral 2396  df-rex 2397  df-reu 2398  df-rab 2400  df-v 2660  df-sbc 2881  df-csb 2974  df-dif 3041  df-un 3043  df-in 3045  df-ss 3052  df-nul 3332  df-pw 3480  df-sn 3501  df-pr 3502  df-op 3504  df-uni 3705  df-iun 3783  df-br 3898  df-opab 3958  df-mpt 3959  df-tr 3995  df-id 4183  df-iord 4256  df-on 4258  df-suc 4261  df-xp 4513  df-rel 4514  df-cnv 4515  df-co 4516  df-dm 4517  df-rn 4518  df-res 4519  df-ima 4520  df-iota 5056  df-fun 5093  df-fn 5094  df-f 5095  df-f1 5096  df-fo 5097  df-f1o 5098  df-fv 5099  df-recs 6168
This theorem is referenced by:  tfrlemiex  6194
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