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Theorem tfrcllemubacc 6373
Description: Lemma for tfrcl 6378. 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 6371 . . . . . . . 8  |-  ( ph  ->  U. B : D --> S )
11 fdm 5383 . . . . . . . 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 6369 . . . . . . . . . 10  |-  ( ph  ->  B  C_  A )
1413unissd 3845 . . . . . . . . 9  |-  ( ph  ->  U. B  C_  U. A
)
155, 3tfrcllemssrecs 6366 . . . . . . . . 9  |-  ( ph  ->  U. A  C_ recs ( G ) )
1614, 15sstrd 3177 . . . . . . . 8  |-  ( ph  ->  U. B  C_ recs ( G ) )
17 dmss 4838 . . . . . . . 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, 18eqsstrrd 3204 . . . . . 6  |-  ( ph  ->  D  C_  dom recs ( G ) )
2019sselda 3167 . . . . 5  |-  ( (
ph  /\  w  e.  D )  ->  w  e.  dom recs ( G ) )
21 eqid 2187 . . . . . 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 6333 . . . . 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 6334 . . . . 5  |-  Fun recs ( G )
2512eleq2d 2257 . . . . . 6  |-  ( ph  ->  ( w  e.  dom  U. B  <->  w  e.  D
) )
2625biimpar 297 . . . . 5  |-  ( (
ph  /\  w  e.  D )  ->  w  e.  dom  U. B )
27 funssfv 5553 . . . . 5  |-  ( ( Fun recs ( G )  /\  U. B  C_ recs ( G )  /\  w  e.  dom  U. B )  ->  (recs ( G ) `  w )  =  ( U. B `  w ) )
2824, 16, 26, 27mp3an2ani 1354 . . . 4  |-  ( (
ph  /\  w  e.  D )  ->  (recs ( G ) `  w
)  =  ( U. B `  w )
)
29 ordelon 4395 . . . . . . . . . 10  |-  ( ( Ord  X  /\  D  e.  X )  ->  D  e.  On )
303, 8, 29syl2anc 411 . . . . . . . . 9  |-  ( ph  ->  D  e.  On )
31 eloni 4387 . . . . . . . . 9  |-  ( D  e.  On  ->  Ord  D )
3230, 31syl 14 . . . . . . . 8  |-  ( ph  ->  Ord  D )
33 ordelss 4391 . . . . . . . 8  |-  ( ( Ord  D  /\  w  e.  D )  ->  w  C_  D )
3432, 33sylan 283 . . . . . . 7  |-  ( (
ph  /\  w  e.  D )  ->  w  C_  D )
3512adantr 276 . . . . . . 7  |-  ( (
ph  /\  w  e.  D )  ->  dom  U. B  =  D )
3634, 35sseqtrrd 3206 . . . . . 6  |-  ( (
ph  /\  w  e.  D )  ->  w  C_ 
dom  U. B )
37 fun2ssres 5271 . . . . . 6  |-  ( ( Fun recs ( G )  /\  U. B  C_ recs ( G )  /\  w  C_ 
dom  U. B )  -> 
(recs ( G )  |`  w )  =  ( U. B  |`  w
) )
3824, 16, 36, 37mp3an2ani 1354 . . . . 5  |-  ( (
ph  /\  w  e.  D )  ->  (recs ( G )  |`  w
)  =  ( U. B  |`  w ) )
3938fveq2d 5531 . . . 4  |-  ( (
ph  /\  w  e.  D )  ->  ( G `  (recs ( G )  |`  w
) )  =  ( G `  ( U. B  |`  w ) ) )
4023, 28, 393eqtr3d 2228 . . 3  |-  ( (
ph  /\  w  e.  D )  ->  ( U. B `  w )  =  ( G `  ( U. B  |`  w
) ) )
4140ralrimiva 2560 . 2  |-  ( ph  ->  A. w  e.  D  ( U. B `  w
)  =  ( G `
 ( U. B  |`  w ) ) )
42 fveq2 5527 . . . 4  |-  ( u  =  w  ->  ( U. B `  u )  =  ( U. B `  w ) )
43 reseq2 4914 . . . . 5  |-  ( u  =  w  ->  ( U. B  |`  u )  =  ( U. B  |`  w ) )
4443fveq2d 5531 . . . 4  |-  ( u  =  w  ->  ( G `  ( U. B  |`  u ) )  =  ( G `  ( U. B  |`  w
) ) )
4542, 44eqeq12d 2202 . . 3  |-  ( u  =  w  ->  (
( U. B `  u )  =  ( G `  ( U. B  |`  u ) )  <-> 
( U. B `  w )  =  ( G `  ( U. B  |`  w ) ) ) )
4645cbvralv 2715 . 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 134 1  |-  ( ph  ->  A. u  e.  D  ( U. B `  u
)  =  ( G `
 ( U. B  |`  u ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    /\ w3a 979    = wceq 1363   E.wex 1502    e. wcel 2158   {cab 2173   A.wral 2465   E.wrex 2466    u. cun 3139    C_ wss 3141   {csn 3604   <.cop 3607   U.cuni 3821   Ord word 4374   Oncon0 4375   suc csuc 4377   dom cdm 4638    |` cres 4640   Fun wfun 5222    Fn wfn 5223   -->wf 5224   ` cfv 5228  recscrecs 6318
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 615  ax-in2 616  ax-io 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-13 2160  ax-14 2161  ax-ext 2169  ax-sep 4133  ax-pow 4186  ax-pr 4221  ax-un 4445  ax-setind 4548
This theorem depends on definitions:  df-bi 117  df-3an 981  df-tru 1366  df-fal 1369  df-nf 1471  df-sb 1773  df-eu 2039  df-mo 2040  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ne 2358  df-ral 2470  df-rex 2471  df-rab 2474  df-v 2751  df-sbc 2975  df-csb 3070  df-dif 3143  df-un 3145  df-in 3147  df-ss 3154  df-nul 3435  df-pw 3589  df-sn 3610  df-pr 3611  df-op 3613  df-uni 3822  df-iun 3900  df-br 4016  df-opab 4077  df-mpt 4078  df-tr 4114  df-id 4305  df-iord 4378  df-on 4380  df-suc 4383  df-xp 4644  df-rel 4645  df-cnv 4646  df-co 4647  df-dm 4648  df-rn 4649  df-res 4650  df-iota 5190  df-fun 5230  df-fn 5231  df-f 5232  df-f1 5233  df-fo 5234  df-f1o 5235  df-fv 5236  df-recs 6319
This theorem is referenced by:  tfrcllemex  6374
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