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Theorem frec0g 6224
Description: The initial value resulting from finite recursive definition generation. (Contributed by Jim Kingdon, 7-May-2020.)
Assertion
Ref Expression
frec0g  |-  ( A  e.  V  ->  (frec ( F ,  A ) `
 (/) )  =  A )

Proof of Theorem frec0g
Dummy variables  g  m  x are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 dm0 4691 . . . . . . . . . 10  |-  dom  (/)  =  (/)
21biantrur 299 . . . . . . . . 9  |-  ( x  e.  A  <->  ( dom  (/)  =  (/)  /\  x  e.  A ) )
3 vex 2644 . . . . . . . . . . . . . . . 16  |-  m  e. 
_V
4 nsuceq0g 4278 . . . . . . . . . . . . . . . 16  |-  ( m  e.  _V  ->  suc  m  =/=  (/) )
53, 4ax-mp 7 . . . . . . . . . . . . . . 15  |-  suc  m  =/=  (/)
65nesymi 2313 . . . . . . . . . . . . . 14  |-  -.  (/)  =  suc  m
71eqeq1i 2107 . . . . . . . . . . . . . 14  |-  ( dom  (/)  =  suc  m  <->  (/)  =  suc  m )
86, 7mtbir 637 . . . . . . . . . . . . 13  |-  -.  dom  (/)  =  suc  m
98intnanr 883 . . . . . . . . . . . 12  |-  -.  ( dom  (/)  =  suc  m  /\  x  e.  ( F `  ( (/) `  m
) ) )
109a1i 9 . . . . . . . . . . 11  |-  ( m  e.  om  ->  -.  ( dom  (/)  =  suc  m  /\  x  e.  ( F `  ( (/) `  m
) ) ) )
1110nrex 2483 . . . . . . . . . 10  |-  -.  E. m  e.  om  ( dom  (/)  =  suc  m  /\  x  e.  ( F `  ( (/) `  m
) ) )
1211biorfi 706 . . . . . . . . 9  |-  ( ( dom  (/)  =  (/)  /\  x  e.  A )  <->  ( ( dom  (/)  =  (/)  /\  x  e.  A )  \/  E. m  e.  om  ( dom  (/)  =  suc  m  /\  x  e.  ( F `  ( (/) `  m
) ) ) ) )
13 orcom 688 . . . . . . . . 9  |-  ( ( ( dom  (/)  =  (/)  /\  x  e.  A )  \/  E. m  e. 
om  ( dom  (/)  =  suc  m  /\  x  e.  ( F `  ( (/) `  m ) ) ) )  <->  ( E. m  e.  om  ( dom  (/)  =  suc  m  /\  x  e.  ( F `  ( (/) `  m ) ) )  \/  ( dom  (/)  =  (/)  /\  x  e.  A ) ) )
142, 12, 133bitri 205 . . . . . . . 8  |-  ( x  e.  A  <->  ( E. m  e.  om  ( dom  (/)  =  suc  m  /\  x  e.  ( F `  ( (/) `  m
) ) )  \/  ( dom  (/)  =  (/)  /\  x  e.  A ) ) )
1514abbii 2215 . . . . . . 7  |-  { x  |  x  e.  A }  =  { x  |  ( E. m  e.  om  ( dom  (/)  =  suc  m  /\  x  e.  ( F `  ( (/) `  m ) ) )  \/  ( dom  (/)  =  (/)  /\  x  e.  A ) ) }
16 abid2 2220 . . . . . . 7  |-  { x  |  x  e.  A }  =  A
1715, 16eqtr3i 2122 . . . . . 6  |-  { x  |  ( E. m  e.  om  ( dom  (/)  =  suc  m  /\  x  e.  ( F `  ( (/) `  m ) ) )  \/  ( dom  (/)  =  (/)  /\  x  e.  A ) ) }  =  A
18 elex 2652 . . . . . 6  |-  ( A  e.  V  ->  A  e.  _V )
1917, 18syl5eqel 2186 . . . . 5  |-  ( A  e.  V  ->  { x  |  ( E. m  e.  om  ( dom  (/)  =  suc  m  /\  x  e.  ( F `  ( (/) `  m ) ) )  \/  ( dom  (/)  =  (/)  /\  x  e.  A ) ) }  e.  _V )
20 0ex 3995 . . . . . . 7  |-  (/)  e.  _V
21 dmeq 4677 . . . . . . . . . . . . 13  |-  ( g  =  (/)  ->  dom  g  =  dom  (/) )
2221eqeq1d 2108 . . . . . . . . . . . 12  |-  ( g  =  (/)  ->  ( dom  g  =  suc  m  <->  dom  (/)  =  suc  m ) )
23 fveq1 5352 . . . . . . . . . . . . . 14  |-  ( g  =  (/)  ->  ( g `
 m )  =  ( (/) `  m ) )
2423fveq2d 5357 . . . . . . . . . . . . 13  |-  ( g  =  (/)  ->  ( F `
 ( g `  m ) )  =  ( F `  ( (/) `  m ) ) )
2524eleq2d 2169 . . . . . . . . . . . 12  |-  ( g  =  (/)  ->  ( x  e.  ( F `  ( g `  m
) )  <->  x  e.  ( F `  ( (/) `  m ) ) ) )
2622, 25anbi12d 460 . . . . . . . . . . 11  |-  ( g  =  (/)  ->  ( ( dom  g  =  suc  m  /\  x  e.  ( F `  ( g `
 m ) ) )  <->  ( dom  (/)  =  suc  m  /\  x  e.  ( F `  ( (/) `  m ) ) ) ) )
2726rexbidv 2397 . . . . . . . . . 10  |-  ( g  =  (/)  ->  ( E. m  e.  om  ( dom  g  =  suc  m  /\  x  e.  ( F `  ( g `
 m ) ) )  <->  E. m  e.  om  ( dom  (/)  =  suc  m  /\  x  e.  ( F `  ( (/) `  m
) ) ) ) )
2821eqeq1d 2108 . . . . . . . . . . 11  |-  ( g  =  (/)  ->  ( dom  g  =  (/)  <->  dom  (/)  =  (/) ) )
2928anbi1d 456 . . . . . . . . . 10  |-  ( g  =  (/)  ->  ( ( dom  g  =  (/)  /\  x  e.  A )  <-> 
( dom  (/)  =  (/)  /\  x  e.  A ) ) )
3027, 29orbi12d 748 . . . . . . . . 9  |-  ( g  =  (/)  ->  ( ( E. m  e.  om  ( dom  g  =  suc  m  /\  x  e.  ( F `  ( g `
 m ) ) )  \/  ( dom  g  =  (/)  /\  x  e.  A ) )  <->  ( E. m  e.  om  ( dom  (/)  =  suc  m  /\  x  e.  ( F `  ( (/) `  m
) ) )  \/  ( dom  (/)  =  (/)  /\  x  e.  A ) ) ) )
3130abbidv 2217 . . . . . . . 8  |-  ( g  =  (/)  ->  { x  |  ( E. m  e.  om  ( dom  g  =  suc  m  /\  x  e.  ( F `  (
g `  m )
) )  \/  ( dom  g  =  (/)  /\  x  e.  A ) ) }  =  { x  |  ( E. m  e. 
om  ( dom  (/)  =  suc  m  /\  x  e.  ( F `  ( (/) `  m ) ) )  \/  ( dom  (/)  =  (/)  /\  x  e.  A ) ) } )
32 eqid 2100 . . . . . . . 8  |-  ( g  e.  _V  |->  { x  |  ( E. m  e.  om  ( dom  g  =  suc  m  /\  x  e.  ( F `  (
g `  m )
) )  \/  ( dom  g  =  (/)  /\  x  e.  A ) ) } )  =  ( g  e.  _V  |->  { x  |  ( E. m  e.  om  ( dom  g  =  suc  m  /\  x  e.  ( F `  (
g `  m )
) )  \/  ( dom  g  =  (/)  /\  x  e.  A ) ) } )
3331, 32fvmptg 5429 . . . . . . 7  |-  ( (
(/)  e.  _V  /\  {
x  |  ( E. m  e.  om  ( dom  (/)  =  suc  m  /\  x  e.  ( F `  ( (/) `  m
) ) )  \/  ( dom  (/)  =  (/)  /\  x  e.  A ) ) }  e.  _V )  ->  ( ( g  e.  _V  |->  { x  |  ( E. m  e.  om  ( dom  g  =  suc  m  /\  x  e.  ( F `  (
g `  m )
) )  \/  ( dom  g  =  (/)  /\  x  e.  A ) ) } ) `  (/) )  =  { x  |  ( E. m  e.  om  ( dom  (/)  =  suc  m  /\  x  e.  ( F `  ( (/) `  m
) ) )  \/  ( dom  (/)  =  (/)  /\  x  e.  A ) ) } )
3420, 33mpan 418 . . . . . 6  |-  ( { x  |  ( E. m  e.  om  ( dom  (/)  =  suc  m  /\  x  e.  ( F `  ( (/) `  m
) ) )  \/  ( dom  (/)  =  (/)  /\  x  e.  A ) ) }  e.  _V  ->  ( ( g  e. 
_V  |->  { x  |  ( E. m  e. 
om  ( dom  g  =  suc  m  /\  x  e.  ( F `  (
g `  m )
) )  \/  ( dom  g  =  (/)  /\  x  e.  A ) ) } ) `  (/) )  =  { x  |  ( E. m  e.  om  ( dom  (/)  =  suc  m  /\  x  e.  ( F `  ( (/) `  m
) ) )  \/  ( dom  (/)  =  (/)  /\  x  e.  A ) ) } )
3534, 17syl6eq 2148 . . . . 5  |-  ( { x  |  ( E. m  e.  om  ( dom  (/)  =  suc  m  /\  x  e.  ( F `  ( (/) `  m
) ) )  \/  ( dom  (/)  =  (/)  /\  x  e.  A ) ) }  e.  _V  ->  ( ( g  e. 
_V  |->  { x  |  ( E. m  e. 
om  ( dom  g  =  suc  m  /\  x  e.  ( F `  (
g `  m )
) )  \/  ( dom  g  =  (/)  /\  x  e.  A ) ) } ) `  (/) )  =  A )
3619, 35syl 14 . . . 4  |-  ( A  e.  V  ->  (
( g  e.  _V  |->  { x  |  ( E. m  e.  om  ( dom  g  =  suc  m  /\  x  e.  ( F `  ( g `
 m ) ) )  \/  ( dom  g  =  (/)  /\  x  e.  A ) ) } ) `  (/) )  =  A )
3736, 18eqeltrd 2176 . . 3  |-  ( A  e.  V  ->  (
( g  e.  _V  |->  { x  |  ( E. m  e.  om  ( dom  g  =  suc  m  /\  x  e.  ( F `  ( g `
 m ) ) )  \/  ( dom  g  =  (/)  /\  x  e.  A ) ) } ) `  (/) )  e. 
_V )
38 df-frec 6218 . . . . . 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 )
3938fveq1i 5354 . . . . 5  |-  (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 ) `  (/) )
40 peano1 4446 . . . . . 6  |-  (/)  e.  om
41 fvres 5377 . . . . . 6  |-  ( (/)  e.  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 ) `  (/) )  =  (recs ( ( g  e. 
_V  |->  { x  |  ( E. m  e. 
om  ( dom  g  =  suc  m  /\  x  e.  ( F `  (
g `  m )
) )  \/  ( dom  g  =  (/)  /\  x  e.  A ) ) } ) ) `  (/) ) )
4240, 41ax-mp 7 . . . . 5  |-  ( (recs ( ( g  e. 
_V  |->  { x  |  ( E. m  e. 
om  ( dom  g  =  suc  m  /\  x  e.  ( F `  (
g `  m )
) )  \/  ( dom  g  =  (/)  /\  x  e.  A ) ) } ) )  |`  om ) `  (/) )  =  (recs ( ( g  e. 
_V  |->  { x  |  ( E. m  e. 
om  ( dom  g  =  suc  m  /\  x  e.  ( F `  (
g `  m )
) )  \/  ( dom  g  =  (/)  /\  x  e.  A ) ) } ) ) `  (/) )
4339, 42eqtri 2120 . . . 4  |-  (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 ) ) } ) ) `  (/) )
44 eqid 2100 . . . . 5  |- recs ( ( g  e.  _V  |->  { x  |  ( E. m  e.  om  ( dom  g  =  suc  m  /\  x  e.  ( F `  ( g `
 m ) ) )  \/  ( dom  g  =  (/)  /\  x  e.  A ) ) } ) )  = recs (
( g  e.  _V  |->  { x  |  ( E. m  e.  om  ( dom  g  =  suc  m  /\  x  e.  ( F `  ( g `
 m ) ) )  \/  ( dom  g  =  (/)  /\  x  e.  A ) ) } ) )
4544tfr0 6150 . . . 4  |-  ( ( ( g  e.  _V  |->  { x  |  ( E. m  e.  om  ( dom  g  =  suc  m  /\  x  e.  ( F `  ( g `
 m ) ) )  \/  ( dom  g  =  (/)  /\  x  e.  A ) ) } ) `  (/) )  e. 
_V  ->  (recs ( ( g  e.  _V  |->  { x  |  ( E. m  e.  om  ( dom  g  =  suc  m  /\  x  e.  ( F `  ( g `
 m ) ) )  \/  ( dom  g  =  (/)  /\  x  e.  A ) ) } ) ) `  (/) )  =  ( ( g  e. 
_V  |->  { x  |  ( E. m  e. 
om  ( dom  g  =  suc  m  /\  x  e.  ( F `  (
g `  m )
) )  \/  ( dom  g  =  (/)  /\  x  e.  A ) ) } ) `  (/) ) )
4643, 45syl5eq 2144 . . 3  |-  ( ( ( g  e.  _V  |->  { x  |  ( E. m  e.  om  ( dom  g  =  suc  m  /\  x  e.  ( F `  ( g `
 m ) ) )  \/  ( dom  g  =  (/)  /\  x  e.  A ) ) } ) `  (/) )  e. 
_V  ->  (frec ( F ,  A ) `  (/) )  =  ( ( g  e.  _V  |->  { x  |  ( E. m  e.  om  ( dom  g  =  suc  m  /\  x  e.  ( F `  ( g `
 m ) ) )  \/  ( dom  g  =  (/)  /\  x  e.  A ) ) } ) `  (/) ) )
4737, 46syl 14 . 2  |-  ( A  e.  V  ->  (frec ( F ,  A ) `
 (/) )  =  ( ( g  e.  _V  |->  { x  |  ( E. m  e.  om  ( dom  g  =  suc  m  /\  x  e.  ( F `  ( g `
 m ) ) )  \/  ( dom  g  =  (/)  /\  x  e.  A ) ) } ) `  (/) ) )
4847, 36eqtrd 2132 1  |-  ( A  e.  V  ->  (frec ( F ,  A ) `
 (/) )  =  A )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 103    \/ wo 670    = wceq 1299    e. wcel 1448   {cab 2086    =/= wne 2267   E.wrex 2376   _Vcvv 2641   (/)c0 3310    |-> cmpt 3929   suc csuc 4225   omcom 4442   dom cdm 4477    |` cres 4479   ` cfv 5059  recscrecs 6131  freccfrec 6217
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 584  ax-in2 585  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-13 1459  ax-14 1460  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082  ax-sep 3986  ax-nul 3994  ax-pow 4038  ax-pr 4069  ax-un 4293  ax-setind 4390
This theorem depends on definitions:  df-bi 116  df-3an 932  df-tru 1302  df-fal 1305  df-nf 1405  df-sb 1704  df-eu 1963  df-mo 1964  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-ne 2268  df-ral 2380  df-rex 2381  df-rab 2384  df-v 2643  df-sbc 2863  df-csb 2956  df-dif 3023  df-un 3025  df-in 3027  df-ss 3034  df-nul 3311  df-pw 3459  df-sn 3480  df-pr 3481  df-op 3483  df-uni 3684  df-int 3719  df-iun 3762  df-br 3876  df-opab 3930  df-mpt 3931  df-tr 3967  df-id 4153  df-iord 4226  df-on 4228  df-suc 4231  df-iom 4443  df-xp 4483  df-rel 4484  df-cnv 4485  df-co 4486  df-dm 4487  df-res 4489  df-iota 5024  df-fun 5061  df-fn 5062  df-fv 5067  df-recs 6132  df-frec 6218
This theorem is referenced by:  frecrdg  6235  frec2uz0d  10013  frec2uzrdg  10023  frecuzrdg0  10027  frecuzrdgg  10030  frecuzrdg0t  10036  seq3val  10072  seqvalcd  10073
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