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Theorem tfrlemibacc 6326
Description: Each element of  B is an acceptable function. Lemma for tfrlemi1 6332. (Contributed by Jim Kingdon, 14-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
tfrlemibacc  |-  ( ph  ->  B  C_  A )
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 tfrlemibacc
StepHypRef Expression
1 tfrlemi1.3 . 2  |-  B  =  { h  |  E. z  e.  x  E. g ( g  Fn  z  /\  g  e.  A  /\  h  =  ( g  u.  { <. z ,  ( F `
 g ) >. } ) ) }
2 simpr3 1005 . . . . . . 7  |-  ( ( ( ph  /\  z  e.  x )  /\  (
g  Fn  z  /\  g  e.  A  /\  h  =  ( g  u.  { <. z ,  ( F `  g )
>. } ) ) )  ->  h  =  ( g  u.  { <. z ,  ( F `  g ) >. } ) )
3 tfrlemisucfn.1 . . . . . . . 8  |-  A  =  { f  |  E. x  e.  On  (
f  Fn  x  /\  A. y  e.  x  ( f `  y )  =  ( F `  ( f  |`  y
) ) ) }
4 tfrlemisucfn.2 . . . . . . . . 9  |-  ( ph  ->  A. x ( Fun 
F  /\  ( F `  x )  e.  _V ) )
54ad2antrr 488 . . . . . . . 8  |-  ( ( ( ph  /\  z  e.  x )  /\  (
g  Fn  z  /\  g  e.  A  /\  h  =  ( g  u.  { <. z ,  ( F `  g )
>. } ) ) )  ->  A. x ( Fun 
F  /\  ( F `  x )  e.  _V ) )
6 tfrlemi1.4 . . . . . . . . . 10  |-  ( ph  ->  x  e.  On )
76ad2antrr 488 . . . . . . . . 9  |-  ( ( ( ph  /\  z  e.  x )  /\  (
g  Fn  z  /\  g  e.  A  /\  h  =  ( g  u.  { <. z ,  ( F `  g )
>. } ) ) )  ->  x  e.  On )
8 simplr 528 . . . . . . . . 9  |-  ( ( ( ph  /\  z  e.  x )  /\  (
g  Fn  z  /\  g  e.  A  /\  h  =  ( g  u.  { <. z ,  ( F `  g )
>. } ) ) )  ->  z  e.  x
)
9 onelon 4384 . . . . . . . . 9  |-  ( ( x  e.  On  /\  z  e.  x )  ->  z  e.  On )
107, 8, 9syl2anc 411 . . . . . . . 8  |-  ( ( ( ph  /\  z  e.  x )  /\  (
g  Fn  z  /\  g  e.  A  /\  h  =  ( g  u.  { <. z ,  ( F `  g )
>. } ) ) )  ->  z  e.  On )
11 simpr1 1003 . . . . . . . 8  |-  ( ( ( ph  /\  z  e.  x )  /\  (
g  Fn  z  /\  g  e.  A  /\  h  =  ( g  u.  { <. z ,  ( F `  g )
>. } ) ) )  ->  g  Fn  z
)
12 simpr2 1004 . . . . . . . 8  |-  ( ( ( ph  /\  z  e.  x )  /\  (
g  Fn  z  /\  g  e.  A  /\  h  =  ( g  u.  { <. z ,  ( F `  g )
>. } ) ) )  ->  g  e.  A
)
133, 5, 10, 11, 12tfrlemisucaccv 6325 . . . . . . 7  |-  ( ( ( ph  /\  z  e.  x )  /\  (
g  Fn  z  /\  g  e.  A  /\  h  =  ( g  u.  { <. z ,  ( F `  g )
>. } ) ) )  ->  ( g  u. 
{ <. z ,  ( F `  g )
>. } )  e.  A
)
142, 13eqeltrd 2254 . . . . . 6  |-  ( ( ( ph  /\  z  e.  x )  /\  (
g  Fn  z  /\  g  e.  A  /\  h  =  ( g  u.  { <. z ,  ( F `  g )
>. } ) ) )  ->  h  e.  A
)
1514ex 115 . . . . 5  |-  ( (
ph  /\  z  e.  x )  ->  (
( g  Fn  z  /\  g  e.  A  /\  h  =  (
g  u.  { <. z ,  ( F `  g ) >. } ) )  ->  h  e.  A ) )
1615exlimdv 1819 . . . 4  |-  ( (
ph  /\  z  e.  x )  ->  ( E. g ( g  Fn  z  /\  g  e.  A  /\  h  =  ( g  u.  { <. z ,  ( F `
 g ) >. } ) )  ->  h  e.  A )
)
1716rexlimdva 2594 . . 3  |-  ( ph  ->  ( E. z  e.  x  E. g ( g  Fn  z  /\  g  e.  A  /\  h  =  ( g  u.  { <. z ,  ( F `  g )
>. } ) )  ->  h  e.  A )
)
1817abssdv 3229 . 2  |-  ( ph  ->  { h  |  E. z  e.  x  E. g ( g  Fn  z  /\  g  e.  A  /\  h  =  ( g  u.  { <. z ,  ( F `
 g ) >. } ) ) } 
C_  A )
191, 18eqsstrid 3201 1  |-  ( ph  ->  B  C_  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    /\ w3a 978   A.wal 1351    = wceq 1353   E.wex 1492    e. wcel 2148   {cab 2163   A.wral 2455   E.wrex 2456   _Vcvv 2737    u. cun 3127    C_ wss 3129   {csn 3592   <.cop 3595   Oncon0 4363    |` cres 4628   Fun wfun 5210    Fn wfn 5211   ` cfv 5216
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 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4121  ax-pow 4174  ax-pr 4209  ax-un 4433  ax-setind 4536
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-v 2739  df-sbc 2963  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-br 4004  df-opab 4065  df-tr 4102  df-id 4293  df-iord 4366  df-on 4368  df-suc 4371  df-xp 4632  df-rel 4633  df-cnv 4634  df-co 4635  df-dm 4636  df-res 4638  df-iota 5178  df-fun 5218  df-fn 5219  df-fv 5224
This theorem is referenced by:  tfrlemibfn  6328  tfrlemiubacc  6330
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