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Theorem tfrcllemubacc 6028
Description: Lemma for tfrcl 6033. The union of  B satisfies the recursion rule. (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
tfrcllemubacc  |-  ( ph  ->  A. u  e.  D  ( U. B `  u
)  =  ( G `
 ( U. B  |`  u ) ) )
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    B, g, h, z    u, B, w    D, h, z   
u, D, w    w, G    h, G, z    u, G    S, g, h, z   
z, X    w, g, ph, y, z
Allowed substitution hints:    ph( u)    A( w, u)    B( x, y, f)    S( w, u)    F( x, y, z, w, u, f, g, h)    G( g)    X( y, w, u, g, h)

Proof of Theorem tfrcllemubacc
Dummy variables  e  t  v are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 tfrcl.f . . . . . . . . 9  |-  F  = recs ( G )
2 tfrcl.g . . . . . . . . 9  |-  ( ph  ->  Fun  G )
3 tfrcl.x . . . . . . . . 9  |-  ( ph  ->  Ord  X )
4 tfrcl.ex . . . . . . . . 9  |-  ( (
ph  /\  x  e.  X  /\  f : x --> S )  ->  ( G `  f )  e.  S )
5 tfrcllemsucfn.1 . . . . . . . . 9  |-  A  =  { f  |  E. x  e.  X  (
f : x --> S  /\  A. y  e.  x  ( f `  y )  =  ( G `  ( f  |`  y
) ) ) }
6 tfrcllembacc.3 . . . . . . . . 9  |-  B  =  { h  |  E. z  e.  D  E. g ( g : z --> S  /\  g  e.  A  /\  h  =  ( g  u. 
{ <. z ,  ( G `  g )
>. } ) ) }
7 tfrcllembacc.u . . . . . . . . 9  |-  ( (
ph  /\  x  e.  U. X )  ->  suc  x  e.  X )
8 tfrcllembacc.4 . . . . . . . . 9  |-  ( ph  ->  D  e.  X )
9 tfrcllembacc.5 . . . . . . . . 9  |-  ( ph  ->  A. z  e.  D  E. g ( g : z --> S  /\  A. w  e.  z  (
g `  w )  =  ( G `  ( g  |`  w
) ) ) )
101, 2, 3, 4, 5, 6, 7, 8, 9tfrcllembfn 6026 . . . . . . . 8  |-  ( ph  ->  U. B : D --> S )
11 fdm 5101 . . . . . . . 8  |-  ( U. B : D --> S  ->  dom  U. B  =  D )
1210, 11syl 14 . . . . . . 7  |-  ( ph  ->  dom  U. B  =  D )
131, 2, 3, 4, 5, 6, 7, 8, 9tfrcllembacc 6024 . . . . . . . . . 10  |-  ( ph  ->  B  C_  A )
1413unissd 3645 . . . . . . . . 9  |-  ( ph  ->  U. B  C_  U. A
)
155, 3tfrcllemssrecs 6021 . . . . . . . . 9  |-  ( ph  ->  U. A  C_ recs ( G ) )
1614, 15sstrd 3018 . . . . . . . 8  |-  ( ph  ->  U. B  C_ recs ( G ) )
17 dmss 4582 . . . . . . . 8  |-  ( U. B  C_ recs ( G )  ->  dom  U. B  C_  dom recs ( G ) )
1816, 17syl 14 . . . . . . 7  |-  ( ph  ->  dom  U. B  C_  dom recs ( G ) )
1912, 18eqsstr3d 3043 . . . . . 6  |-  ( ph  ->  D  C_  dom recs ( G ) )
2019sselda 3008 . . . . 5  |-  ( (
ph  /\  w  e.  D )  ->  w  e.  dom recs ( G ) )
21 eqid 2083 . . . . . 6  |-  { e  |  E. v  e.  On  ( e  Fn  v  /\  A. t  e.  v  ( e `  t )  =  ( G `  ( e  |`  t ) ) ) }  =  { e  |  E. v  e.  On  ( e  Fn  v  /\  A. t  e.  v  ( e `  t )  =  ( G `  ( e  |`  t ) ) ) }
2221tfrlem9 5988 . . . . 5  |-  ( w  e.  dom recs ( G
)  ->  (recs ( G ) `  w
)  =  ( G `
 (recs ( G )  |`  w )
) )
2320, 22syl 14 . . . 4  |-  ( (
ph  /\  w  e.  D )  ->  (recs ( G ) `  w
)  =  ( G `
 (recs ( G )  |`  w )
) )
24 tfrfun 5989 . . . . 5  |-  Fun recs ( G )
2512eleq2d 2152 . . . . . 6  |-  ( ph  ->  ( w  e.  dom  U. B  <->  w  e.  D
) )
2625biimpar 291 . . . . 5  |-  ( (
ph  /\  w  e.  D )  ->  w  e.  dom  U. B )
27 funssfv 5251 . . . . 5  |-  ( ( Fun recs ( G )  /\  U. B  C_ recs ( G )  /\  w  e.  dom  U. B )  ->  (recs ( G ) `  w )  =  ( U. B `  w ) )
2824, 16, 26, 27mp3an2ani 1276 . . . 4  |-  ( (
ph  /\  w  e.  D )  ->  (recs ( G ) `  w
)  =  ( U. B `  w )
)
29 ordelon 4166 . . . . . . . . . 10  |-  ( ( Ord  X  /\  D  e.  X )  ->  D  e.  On )
303, 8, 29syl2anc 403 . . . . . . . . 9  |-  ( ph  ->  D  e.  On )
31 eloni 4158 . . . . . . . . 9  |-  ( D  e.  On  ->  Ord  D )
3230, 31syl 14 . . . . . . . 8  |-  ( ph  ->  Ord  D )
33 ordelss 4162 . . . . . . . 8  |-  ( ( Ord  D  /\  w  e.  D )  ->  w  C_  D )
3432, 33sylan 277 . . . . . . 7  |-  ( (
ph  /\  w  e.  D )  ->  w  C_  D )
3512adantr 270 . . . . . . 7  |-  ( (
ph  /\  w  e.  D )  ->  dom  U. B  =  D )
3634, 35sseqtr4d 3045 . . . . . 6  |-  ( (
ph  /\  w  e.  D )  ->  w  C_ 
dom  U. B )
37 fun2ssres 4993 . . . . . 6  |-  ( ( Fun recs ( G )  /\  U. B  C_ recs ( G )  /\  w  C_ 
dom  U. B )  -> 
(recs ( G )  |`  w )  =  ( U. B  |`  w
) )
3824, 16, 36, 37mp3an2ani 1276 . . . . 5  |-  ( (
ph  /\  w  e.  D )  ->  (recs ( G )  |`  w
)  =  ( U. B  |`  w ) )
3938fveq2d 5233 . . . 4  |-  ( (
ph  /\  w  e.  D )  ->  ( G `  (recs ( G )  |`  w
) )  =  ( G `  ( U. B  |`  w ) ) )
4023, 28, 393eqtr3d 2123 . . 3  |-  ( (
ph  /\  w  e.  D )  ->  ( U. B `  w )  =  ( G `  ( U. B  |`  w
) ) )
4140ralrimiva 2439 . 2  |-  ( ph  ->  A. w  e.  D  ( U. B `  w
)  =  ( G `
 ( U. B  |`  w ) ) )
42 fveq2 5229 . . . 4  |-  ( u  =  w  ->  ( U. B `  u )  =  ( U. B `  w ) )
43 reseq2 4655 . . . . 5  |-  ( u  =  w  ->  ( U. B  |`  u )  =  ( U. B  |`  w ) )
4443fveq2d 5233 . . . 4  |-  ( u  =  w  ->  ( G `  ( U. B  |`  u ) )  =  ( G `  ( U. B  |`  w
) ) )
4542, 44eqeq12d 2097 . . 3  |-  ( u  =  w  ->  (
( U. B `  u )  =  ( G `  ( U. B  |`  u ) )  <-> 
( U. B `  w )  =  ( G `  ( U. B  |`  w ) ) ) )
4645cbvralv 2582 . 2  |-  ( A. u  e.  D  ( U. B `  u )  =  ( G `  ( U. B  |`  u
) )  <->  A. w  e.  D  ( U. B `  w )  =  ( G `  ( U. B  |`  w
) ) )
4741, 46sylibr 132 1  |-  ( ph  ->  A. u  e.  D  ( U. B `  u
)  =  ( G `
 ( U. B  |`  u ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    /\ w3a 920    = wceq 1285   E.wex 1422    e. wcel 1434   {cab 2069   A.wral 2353   E.wrex 2354    u. cun 2980    C_ wss 2982   {csn 3416   <.cop 3419   U.cuni 3621   Ord word 4145   Oncon0 4146   suc csuc 4148   dom cdm 4391    |` cres 4393   Fun wfun 4946    Fn wfn 4947   -->wf 4948   ` cfv 4952  recscrecs 5973
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 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-sep 3916  ax-pow 3968  ax-pr 3992  ax-un 4216  ax-setind 4308
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ne 2250  df-ral 2358  df-rex 2359  df-rab 2362  df-v 2612  df-sbc 2825  df-csb 2918  df-dif 2984  df-un 2986  df-in 2988  df-ss 2995  df-nul 3268  df-pw 3402  df-sn 3422  df-pr 3423  df-op 3425  df-uni 3622  df-iun 3700  df-br 3806  df-opab 3860  df-mpt 3861  df-tr 3896  df-id 4076  df-iord 4149  df-on 4151  df-suc 4154  df-xp 4397  df-rel 4398  df-cnv 4399  df-co 4400  df-dm 4401  df-rn 4402  df-res 4403  df-iota 4917  df-fun 4954  df-fn 4955  df-f 4956  df-f1 4957  df-fo 4958  df-f1o 4959  df-fv 4960  df-recs 5974
This theorem is referenced by:  tfrcllemex  6029
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