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Theorem tfrcllembacc 6220
Description: Lemma for tfrcl 6229. 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 974 . . . . . . 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 479 . . . . . . . 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 479 . . . . . . . 8  |-  ( ( ( ph  /\  z  e.  D )  /\  (
g : z --> S  /\  g  e.  A  /\  h  =  (
g  u.  { <. z ,  ( G `  g ) >. } ) ) )  ->  Ord  X )
8 simp1ll 1029 . . . . . . . . 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 1247 . . . . . . . 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 479 . . . . . . . 8  |-  ( ( ( ph  /\  z  e.  D )  /\  (
g : z --> S  /\  g  e.  A  /\  h  =  (
g  u.  { <. z ,  ( G `  g ) >. } ) ) )  ->  D  e.  X )
14 simplr 504 . . . . . . . 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 468 . . . . . . . . 9  |-  ( ( ( ph  /\  z  e.  D )  /\  x  e.  U. X )  ->  suc  x  e.  X )
1716adantlr 468 . . . . . . . 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 972 . . . . . . . 8  |-  ( ( ( ph  /\  z  e.  D )  /\  (
g : z --> S  /\  g  e.  A  /\  h  =  (
g  u.  { <. z ,  ( G `  g ) >. } ) ) )  ->  g : z --> S )
19 simpr2 973 . . . . . . . 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 6219 . . . . . . 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 2194 . . . . . 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 1775 . . . 4  |-  ( (
ph  /\  z  e.  D )  ->  ( E. g ( g : z --> S  /\  g  e.  A  /\  h  =  ( g  u. 
{ <. z ,  ( G `  g )
>. } ) )  ->  h  e.  A )
)
2423rexlimdva 2526 . . 3  |-  ( ph  ->  ( E. z  e.  D  E. g ( g : z --> S  /\  g  e.  A  /\  h  =  (
g  u.  { <. z ,  ( G `  g ) >. } ) )  ->  h  e.  A ) )
2524abssdv 3141 . 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 3113 1  |-  ( ph  ->  B  C_  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    /\ w3a 947    = wceq 1316   E.wex 1453    e. wcel 1465   {cab 2103   A.wral 2393   E.wrex 2394    u. cun 3039    C_ wss 3041   {csn 3497   <.cop 3500   U.cuni 3706   Ord word 4254   suc csuc 4257    |` cres 4511   Fun wfun 5087   -->wf 5089   ` cfv 5093  recscrecs 6169
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 588  ax-in2 589  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-13 1476  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016  ax-pow 4068  ax-pr 4101  ax-un 4325  ax-setind 4422
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-fal 1322  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ne 2286  df-ral 2398  df-rex 2399  df-v 2662  df-sbc 2883  df-dif 3043  df-un 3045  df-in 3047  df-ss 3054  df-nul 3334  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-uni 3707  df-br 3900  df-opab 3960  df-tr 3997  df-id 4185  df-iord 4258  df-on 4260  df-suc 4263  df-xp 4515  df-rel 4516  df-cnv 4517  df-co 4518  df-dm 4519  df-rn 4520  df-res 4521  df-iota 5058  df-fun 5095  df-fn 5096  df-f 5097  df-f1 5098  df-fo 5099  df-f1o 5100  df-fv 5101
This theorem is referenced by:  tfrcllembfn  6222  tfrcllemubacc  6224
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