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Theorem tfrcllembacc 6304
Description: Lemma for tfrcl 6313. Each element of  B is an acceptable function. (Contributed by Jim Kingdon, 25-Mar-2022.)
Hypotheses
Ref Expression
tfrcl.f  |-  F  = recs ( G )
tfrcl.g  |-  ( ph  ->  Fun  G )
tfrcl.x  |-  ( ph  ->  Ord  X )
tfrcl.ex  |-  ( (
ph  /\  x  e.  X  /\  f : x --> S )  ->  ( G `  f )  e.  S )
tfrcllemsucfn.1  |-  A  =  { f  |  E. x  e.  X  (
f : x --> S  /\  A. y  e.  x  ( f `  y )  =  ( G `  ( f  |`  y
) ) ) }
tfrcllembacc.3  |-  B  =  { h  |  E. z  e.  D  E. g ( g : z --> S  /\  g  e.  A  /\  h  =  ( g  u. 
{ <. z ,  ( G `  g )
>. } ) ) }
tfrcllembacc.u  |-  ( (
ph  /\  x  e.  U. X )  ->  suc  x  e.  X )
tfrcllembacc.4  |-  ( ph  ->  D  e.  X )
tfrcllembacc.5  |-  ( ph  ->  A. z  e.  D  E. g ( g : z --> S  /\  A. w  e.  z  (
g `  w )  =  ( G `  ( g  |`  w
) ) ) )
Assertion
Ref Expression
tfrcllembacc  |-  ( ph  ->  B  C_  A )
Distinct variable groups:    A, f, g, h, x, y, z    D, f, g, x, y   
f, G, x, y    S, f, x, y    f, X, x    ph, f, g, h, x, y, z
Allowed substitution hints:    ph( w)    A( w)    B( x, y, z, w, f, g, h)    D( z, w, h)    S( z, w, g, h)    F( x, y, z, w, f, g, h)    G( z, w, g, h)    X( y, z, w, g, h)

Proof of Theorem tfrcllembacc
StepHypRef Expression
1 tfrcllembacc.3 . 2  |-  B  =  { h  |  E. z  e.  D  E. g ( g : z --> S  /\  g  e.  A  /\  h  =  ( g  u. 
{ <. z ,  ( G `  g )
>. } ) ) }
2 simpr3 990 . . . . . . 7  |-  ( ( ( ph  /\  z  e.  D )  /\  (
g : z --> S  /\  g  e.  A  /\  h  =  (
g  u.  { <. z ,  ( G `  g ) >. } ) ) )  ->  h  =  ( g  u. 
{ <. z ,  ( G `  g )
>. } ) )
3 tfrcl.f . . . . . . . 8  |-  F  = recs ( G )
4 tfrcl.g . . . . . . . . 9  |-  ( ph  ->  Fun  G )
54ad2antrr 480 . . . . . . . 8  |-  ( ( ( ph  /\  z  e.  D )  /\  (
g : z --> S  /\  g  e.  A  /\  h  =  (
g  u.  { <. z ,  ( G `  g ) >. } ) ) )  ->  Fun  G )
6 tfrcl.x . . . . . . . . 9  |-  ( ph  ->  Ord  X )
76ad2antrr 480 . . . . . . . 8  |-  ( ( ( ph  /\  z  e.  D )  /\  (
g : z --> S  /\  g  e.  A  /\  h  =  (
g  u.  { <. z ,  ( G `  g ) >. } ) ) )  ->  Ord  X )
8 simp1ll 1045 . . . . . . . . 9  |-  ( ( ( ( ph  /\  z  e.  D )  /\  ( g : z --> S  /\  g  e.  A  /\  h  =  ( g  u.  { <. z ,  ( G `
 g ) >. } ) ) )  /\  x  e.  X  /\  f : x --> S )  ->  ph )
9 tfrcl.ex . . . . . . . . 9  |-  ( (
ph  /\  x  e.  X  /\  f : x --> S )  ->  ( G `  f )  e.  S )
108, 9syld3an1 1266 . . . . . . . 8  |-  ( ( ( ( ph  /\  z  e.  D )  /\  ( g : z --> S  /\  g  e.  A  /\  h  =  ( g  u.  { <. z ,  ( G `
 g ) >. } ) ) )  /\  x  e.  X  /\  f : x --> S )  ->  ( G `  f )  e.  S
)
11 tfrcllemsucfn.1 . . . . . . . 8  |-  A  =  { f  |  E. x  e.  X  (
f : x --> S  /\  A. y  e.  x  ( f `  y )  =  ( G `  ( f  |`  y
) ) ) }
12 tfrcllembacc.4 . . . . . . . . 9  |-  ( ph  ->  D  e.  X )
1312ad2antrr 480 . . . . . . . 8  |-  ( ( ( ph  /\  z  e.  D )  /\  (
g : z --> S  /\  g  e.  A  /\  h  =  (
g  u.  { <. z ,  ( G `  g ) >. } ) ) )  ->  D  e.  X )
14 simplr 520 . . . . . . . 8  |-  ( ( ( ph  /\  z  e.  D )  /\  (
g : z --> S  /\  g  e.  A  /\  h  =  (
g  u.  { <. z ,  ( G `  g ) >. } ) ) )  ->  z  e.  D )
15 tfrcllembacc.u . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  U. X )  ->  suc  x  e.  X )
1615adantlr 469 . . . . . . . . 9  |-  ( ( ( ph  /\  z  e.  D )  /\  x  e.  U. X )  ->  suc  x  e.  X )
1716adantlr 469 . . . . . . . 8  |-  ( ( ( ( ph  /\  z  e.  D )  /\  ( g : z --> S  /\  g  e.  A  /\  h  =  ( g  u.  { <. z ,  ( G `
 g ) >. } ) ) )  /\  x  e.  U. X )  ->  suc  x  e.  X )
18 simpr1 988 . . . . . . . 8  |-  ( ( ( ph  /\  z  e.  D )  /\  (
g : z --> S  /\  g  e.  A  /\  h  =  (
g  u.  { <. z ,  ( G `  g ) >. } ) ) )  ->  g : z --> S )
19 simpr2 989 . . . . . . . 8  |-  ( ( ( ph  /\  z  e.  D )  /\  (
g : z --> S  /\  g  e.  A  /\  h  =  (
g  u.  { <. z ,  ( G `  g ) >. } ) ) )  ->  g  e.  A )
203, 5, 7, 10, 11, 13, 14, 17, 18, 19tfrcllemsucaccv 6303 . . . . . . 7  |-  ( ( ( ph  /\  z  e.  D )  /\  (
g : z --> S  /\  g  e.  A  /\  h  =  (
g  u.  { <. z ,  ( G `  g ) >. } ) ) )  ->  (
g  u.  { <. z ,  ( G `  g ) >. } )  e.  A )
212, 20eqeltrd 2234 . . . . . 6  |-  ( ( ( ph  /\  z  e.  D )  /\  (
g : z --> S  /\  g  e.  A  /\  h  =  (
g  u.  { <. z ,  ( G `  g ) >. } ) ) )  ->  h  e.  A )
2221ex 114 . . . . 5  |-  ( (
ph  /\  z  e.  D )  ->  (
( g : z --> S  /\  g  e.  A  /\  h  =  ( g  u.  { <. z ,  ( G `
 g ) >. } ) )  ->  h  e.  A )
)
2322exlimdv 1799 . . . 4  |-  ( (
ph  /\  z  e.  D )  ->  ( E. g ( g : z --> S  /\  g  e.  A  /\  h  =  ( g  u. 
{ <. z ,  ( G `  g )
>. } ) )  ->  h  e.  A )
)
2423rexlimdva 2574 . . 3  |-  ( ph  ->  ( E. z  e.  D  E. g ( g : z --> S  /\  g  e.  A  /\  h  =  (
g  u.  { <. z ,  ( G `  g ) >. } ) )  ->  h  e.  A ) )
2524abssdv 3202 . 2  |-  ( ph  ->  { h  |  E. z  e.  D  E. g ( g : z --> S  /\  g  e.  A  /\  h  =  ( g  u. 
{ <. z ,  ( G `  g )
>. } ) ) } 
C_  A )
261, 25eqsstrid 3174 1  |-  ( ph  ->  B  C_  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    /\ w3a 963    = wceq 1335   E.wex 1472    e. wcel 2128   {cab 2143   A.wral 2435   E.wrex 2436    u. cun 3100    C_ wss 3102   {csn 3561   <.cop 3564   U.cuni 3774   Ord word 4324   suc csuc 4327    |` cres 4590   Fun wfun 5166   -->wf 5168   ` cfv 5172  recscrecs 6253
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 604  ax-in2 605  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-13 2130  ax-14 2131  ax-ext 2139  ax-sep 4084  ax-pow 4137  ax-pr 4171  ax-un 4395  ax-setind 4498
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-fal 1341  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ne 2328  df-ral 2440  df-rex 2441  df-v 2714  df-sbc 2938  df-dif 3104  df-un 3106  df-in 3108  df-ss 3115  df-nul 3396  df-pw 3546  df-sn 3567  df-pr 3568  df-op 3570  df-uni 3775  df-br 3968  df-opab 4028  df-tr 4065  df-id 4255  df-iord 4328  df-on 4330  df-suc 4333  df-xp 4594  df-rel 4595  df-cnv 4596  df-co 4597  df-dm 4598  df-rn 4599  df-res 4600  df-iota 5137  df-fun 5174  df-fn 5175  df-f 5176  df-f1 5177  df-fo 5178  df-f1o 5179  df-fv 5180
This theorem is referenced by:  tfrcllembfn  6306  tfrcllemubacc  6308
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