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Theorem frecsuclem1 6018
Description: Lemma for frecsuc 6022. (Contributed by Jim Kingdon, 13-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
frecsuclem1  |-  ( ( A. z ( F `
 z )  e. 
_V  /\  A  e.  V  /\  B  e.  om )  ->  (frec ( F ,  A ) `  suc  B )  =  ( G `  (recs ( G )  |`  suc  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 frecsuclem1
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 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 )
2 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 ) ) } )
3 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 ) ) } ) ) )
42, 3ax-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 ) ) } ) )
54reseq1i 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 )
61, 5eqtr4i 2079 . . . . 5  |- frec ( F ,  A )  =  (recs ( G )  |`  om )
76fveq1i 5207 . . . 4  |-  (frec ( F ,  A ) `
 suc  B )  =  ( (recs ( G )  |`  om ) `  suc  B )
8 peano2 4346 . . . . 5  |-  ( B  e.  om  ->  suc  B  e.  om )
9 fvres 5226 . . . . 5  |-  ( suc 
B  e.  om  ->  ( (recs ( G )  |`  om ) `  suc  B )  =  (recs ( G ) `  suc  B ) )
108, 9syl 14 . . . 4  |-  ( B  e.  om  ->  (
(recs ( G )  |`  om ) `  suc  B )  =  (recs ( G ) `  suc  B ) )
117, 10syl5eq 2100 . . 3  |-  ( B  e.  om  ->  (frec ( F ,  A ) `
 suc  B )  =  (recs ( G ) `
 suc  B )
)
12113ad2ant3 938 . 2  |-  ( ( A. z ( F `
 z )  e. 
_V  /\  A  e.  V  /\  B  e.  om )  ->  (frec ( F ,  A ) `  suc  B )  =  (recs ( G ) `  suc  B ) )
13 nnon 4360 . . . . 5  |-  ( suc 
B  e.  om  ->  suc 
B  e.  On )
148, 13syl 14 . . . 4  |-  ( B  e.  om  ->  suc  B  e.  On )
15 eqid 2056 . . . . 5  |- recs ( G )  = recs ( G )
162frectfr 6016 . . . . 5  |-  ( ( A. z ( F `
 z )  e. 
_V  /\  A  e.  V )  ->  A. y
( Fun  G  /\  ( G `  y )  e.  _V ) )
1715, 16tfri2d 5981 . . . 4  |-  ( ( ( A. z ( F `  z )  e.  _V  /\  A  e.  V )  /\  suc  B  e.  On )  -> 
(recs ( G ) `
 suc  B )  =  ( G `  (recs ( G )  |`  suc  B ) ) )
1814, 17sylan2 274 . . 3  |-  ( ( ( A. z ( F `  z )  e.  _V  /\  A  e.  V )  /\  B  e.  om )  ->  (recs ( G ) `  suc  B )  =  ( G `
 (recs ( G )  |`  suc  B ) ) )
19183impa 1110 . 2  |-  ( ( A. z ( F `
 z )  e. 
_V  /\  A  e.  V  /\  B  e.  om )  ->  (recs ( G ) `  suc  B
)  =  ( G `
 (recs ( G )  |`  suc  B ) ) )
2012, 19eqtrd 2088 1  |-  ( ( A. z ( F `
 z )  e. 
_V  /\  A  e.  V  /\  B  e.  om )  ->  (frec ( F ,  A ) `  suc  B )  =  ( G `  (recs ( G )  |`  suc  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   Oncon0 4128   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-in1 554  ax-in2 555  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-13 1420  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-coll 3900  ax-sep 3903  ax-nul 3911  ax-pow 3955  ax-pr 3972  ax-un 4198  ax-setind 4290  ax-iinf 4339
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-fal 1265  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ne 2221  df-ral 2328  df-rex 2329  df-reu 2330  df-rab 2332  df-v 2576  df-sbc 2788  df-csb 2881  df-dif 2948  df-un 2950  df-in 2952  df-ss 2959  df-nul 3253  df-pw 3389  df-sn 3409  df-pr 3410  df-op 3412  df-uni 3609  df-int 3644  df-iun 3687  df-br 3793  df-opab 3847  df-mpt 3848  df-tr 3883  df-id 4058  df-iord 4131  df-on 4133  df-suc 4136  df-iom 4342  df-xp 4379  df-rel 4380  df-cnv 4381  df-co 4382  df-dm 4383  df-rn 4384  df-res 4385  df-ima 4386  df-iota 4895  df-fun 4932  df-fn 4933  df-f 4934  df-f1 4935  df-fo 4936  df-f1o 4937  df-fv 4938  df-recs 5951  df-frec 6009
This theorem is referenced by:  frecsuclem3  6021
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