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Theorem tfrlemibacc 6073
Description: Each element of  B is an acceptable function. Lemma for tfrlemi1 6079. (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 951 . . . . . . 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 472 . . . . . . . 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 472 . . . . . . . . 9  |-  ( ( ( ph  /\  z  e.  x )  /\  (
g  Fn  z  /\  g  e.  A  /\  h  =  ( g  u.  { <. z ,  ( F `  g )
>. } ) ) )  ->  x  e.  On )
8 simplr 497 . . . . . . . . 9  |-  ( ( ( ph  /\  z  e.  x )  /\  (
g  Fn  z  /\  g  e.  A  /\  h  =  ( g  u.  { <. z ,  ( F `  g )
>. } ) ) )  ->  z  e.  x
)
9 onelon 4202 . . . . . . . . 9  |-  ( ( x  e.  On  /\  z  e.  x )  ->  z  e.  On )
107, 8, 9syl2anc 403 . . . . . . . 8  |-  ( ( ( ph  /\  z  e.  x )  /\  (
g  Fn  z  /\  g  e.  A  /\  h  =  ( g  u.  { <. z ,  ( F `  g )
>. } ) ) )  ->  z  e.  On )
11 simpr1 949 . . . . . . . 8  |-  ( ( ( ph  /\  z  e.  x )  /\  (
g  Fn  z  /\  g  e.  A  /\  h  =  ( g  u.  { <. z ,  ( F `  g )
>. } ) ) )  ->  g  Fn  z
)
12 simpr2 950 . . . . . . . 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 6072 . . . . . . 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 2164 . . . . . 6  |-  ( ( ( ph  /\  z  e.  x )  /\  (
g  Fn  z  /\  g  e.  A  /\  h  =  ( g  u.  { <. z ,  ( F `  g )
>. } ) ) )  ->  h  e.  A
)
1514ex 113 . . . . 5  |-  ( (
ph  /\  z  e.  x )  ->  (
( g  Fn  z  /\  g  e.  A  /\  h  =  (
g  u.  { <. z ,  ( F `  g ) >. } ) )  ->  h  e.  A ) )
1615exlimdv 1747 . . . 4  |-  ( (
ph  /\  z  e.  x )  ->  ( E. g ( g  Fn  z  /\  g  e.  A  /\  h  =  ( g  u.  { <. z ,  ( F `
 g ) >. } ) )  ->  h  e.  A )
)
1716rexlimdva 2489 . . 3  |-  ( ph  ->  ( E. z  e.  x  E. g ( g  Fn  z  /\  g  e.  A  /\  h  =  ( g  u.  { <. z ,  ( F `  g )
>. } ) )  ->  h  e.  A )
)
1817abssdv 3093 . 2  |-  ( ph  ->  { h  |  E. z  e.  x  E. g ( g  Fn  z  /\  g  e.  A  /\  h  =  ( g  u.  { <. z ,  ( F `
 g ) >. } ) ) } 
C_  A )
191, 18syl5eqss 3068 1  |-  ( ph  ->  B  C_  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    /\ w3a 924   A.wal 1287    = wceq 1289   E.wex 1426    e. wcel 1438   {cab 2074   A.wral 2359   E.wrex 2360   _Vcvv 2619    u. cun 2995    C_ wss 2997   {csn 3441   <.cop 3444   Oncon0 4181    |` cres 4430   Fun wfun 4996    Fn wfn 4997   ` cfv 5002
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 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3949  ax-pow 4001  ax-pr 4027  ax-un 4251  ax-setind 4343
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ne 2256  df-ral 2364  df-rex 2365  df-v 2621  df-sbc 2839  df-dif 2999  df-un 3001  df-in 3003  df-ss 3010  df-nul 3285  df-pw 3427  df-sn 3447  df-pr 3448  df-op 3450  df-uni 3649  df-br 3838  df-opab 3892  df-tr 3929  df-id 4111  df-iord 4184  df-on 4186  df-suc 4189  df-xp 4434  df-rel 4435  df-cnv 4436  df-co 4437  df-dm 4438  df-res 4440  df-iota 4967  df-fun 5004  df-fn 5005  df-fv 5010
This theorem is referenced by:  tfrlemibfn  6075  tfrlemiubacc  6077
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