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Theorem frecsuclem2 6020
Description: Lemma for frecsuc 6022. (Contributed by Jim Kingdon, 15-Aug-2019.)
Hypothesis
Ref Expression
frecsuclem1.h  |-  G  =  ( g  e.  _V  |->  { x  |  ( E. m  e.  om  ( dom  g  =  suc  m  /\  x  e.  ( F `  ( g `
 m ) ) )  \/  ( dom  g  =  (/)  /\  x  e.  A ) ) } )
Assertion
Ref Expression
frecsuclem2  |-  ( ( A. z ( F `
 z )  e. 
_V  /\  A  e.  V  /\  B  e.  om )  ->  ( (recs ( G )  |`  suc  B
) `  B )  =  (frec ( F ,  A ) `  B
) )
Distinct variable groups:    A, g, m, x, z    B, g, m, x, z    g, F, m, x, z    g, G, m, x, z    g, V, m, x
Allowed substitution hint:    V( z)

Proof of Theorem frecsuclem2
StepHypRef Expression
1 sucidg 4181 . . . 4  |-  ( B  e.  om  ->  B  e.  suc  B )
2 fvres 5226 . . . 4  |-  ( B  e.  suc  B  -> 
( (recs ( G )  |`  suc  B ) `
 B )  =  (recs ( G ) `
 B ) )
31, 2syl 14 . . 3  |-  ( B  e.  om  ->  (
(recs ( G )  |`  suc  B ) `  B )  =  (recs ( G ) `  B ) )
4 df-frec 6009 . . . . . 6  |- frec ( F ,  A )  =  (recs ( ( g  e.  _V  |->  { x  |  ( E. m  e.  om  ( dom  g  =  suc  m  /\  x  e.  ( F `  (
g `  m )
) )  \/  ( dom  g  =  (/)  /\  x  e.  A ) ) } ) )  |`  om )
5 frecsuclem1.h . . . . . . . 8  |-  G  =  ( g  e.  _V  |->  { x  |  ( E. m  e.  om  ( dom  g  =  suc  m  /\  x  e.  ( F `  ( g `
 m ) ) )  \/  ( dom  g  =  (/)  /\  x  e.  A ) ) } )
6 recseq 5952 . . . . . . . 8  |-  ( G  =  ( g  e. 
_V  |->  { x  |  ( E. m  e. 
om  ( dom  g  =  suc  m  /\  x  e.  ( F `  (
g `  m )
) )  \/  ( dom  g  =  (/)  /\  x  e.  A ) ) } )  -> recs ( G
)  = recs ( ( g  e.  _V  |->  { x  |  ( E. m  e.  om  ( dom  g  =  suc  m  /\  x  e.  ( F `  ( g `
 m ) ) )  \/  ( dom  g  =  (/)  /\  x  e.  A ) ) } ) ) )
75, 6ax-mp 7 . . . . . . 7  |- recs ( G )  = recs ( ( g  e.  _V  |->  { x  |  ( E. m  e.  om  ( dom  g  =  suc  m  /\  x  e.  ( F `  ( g `
 m ) ) )  \/  ( dom  g  =  (/)  /\  x  e.  A ) ) } ) )
87reseq1i 4636 . . . . . 6  |-  (recs ( G )  |`  om )  =  (recs ( ( g  e.  _V  |->  { x  |  ( E. m  e.  om  ( dom  g  =  suc  m  /\  x  e.  ( F `  (
g `  m )
) )  \/  ( dom  g  =  (/)  /\  x  e.  A ) ) } ) )  |`  om )
94, 8eqtr4i 2079 . . . . 5  |- frec ( F ,  A )  =  (recs ( G )  |`  om )
109fveq1i 5207 . . . 4  |-  (frec ( F ,  A ) `
 B )  =  ( (recs ( G )  |`  om ) `  B )
11 fvres 5226 . . . 4  |-  ( B  e.  om  ->  (
(recs ( G )  |`  om ) `  B
)  =  (recs ( G ) `  B
) )
1210, 11syl5eq 2100 . . 3  |-  ( B  e.  om  ->  (frec ( F ,  A ) `
 B )  =  (recs ( G ) `
 B ) )
133, 12eqtr4d 2091 . 2  |-  ( B  e.  om  ->  (
(recs ( G )  |`  suc  B ) `  B )  =  (frec ( F ,  A
) `  B )
)
14133ad2ant3 938 1  |-  ( ( A. z ( F `
 z )  e. 
_V  /\  A  e.  V  /\  B  e.  om )  ->  ( (recs ( G )  |`  suc  B
) `  B )  =  (frec ( F ,  A ) `  B
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 101    \/ wo 639    /\ w3a 896   A.wal 1257    = wceq 1259    e. wcel 1409   {cab 2042   E.wrex 2324   _Vcvv 2574   (/)c0 3252    |-> cmpt 3846   suc csuc 4130   omcom 4341   dom cdm 4373    |` cres 4375   ` cfv 4930  recscrecs 5950  freccfrec 6008
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3903  ax-pow 3955  ax-pr 3972
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-rex 2329  df-v 2576  df-un 2950  df-in 2952  df-ss 2959  df-pw 3389  df-sn 3409  df-pr 3410  df-op 3412  df-uni 3609  df-br 3793  df-opab 3847  df-suc 4136  df-xp 4379  df-res 4385  df-iota 4895  df-fv 4938  df-recs 5951  df-frec 6009
This theorem is referenced by:  frecsuclem3  6021
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