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Theorem frec0g 6294
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 4753 . . . . . . . . . 10  |-  dom  (/)  =  (/)
21biantrur 301 . . . . . . . . 9  |-  ( x  e.  A  <->  ( dom  (/)  =  (/)  /\  x  e.  A ) )
3 vex 2689 . . . . . . . . . . . . . . . 16  |-  m  e. 
_V
4 nsuceq0g 4340 . . . . . . . . . . . . . . . 16  |-  ( m  e.  _V  ->  suc  m  =/=  (/) )
53, 4ax-mp 5 . . . . . . . . . . . . . . 15  |-  suc  m  =/=  (/)
65nesymi 2354 . . . . . . . . . . . . . 14  |-  -.  (/)  =  suc  m
71eqeq1i 2147 . . . . . . . . . . . . . 14  |-  ( dom  (/)  =  suc  m  <->  (/)  =  suc  m )
86, 7mtbir 660 . . . . . . . . . . . . 13  |-  -.  dom  (/)  =  suc  m
98intnanr 915 . . . . . . . . . . . 12  |-  -.  ( dom  (/)  =  suc  m  /\  x  e.  ( F `  ( (/) `  m
) ) )
109a1i 9 . . . . . . . . . . 11  |-  ( m  e.  om  ->  -.  ( dom  (/)  =  suc  m  /\  x  e.  ( F `  ( (/) `  m
) ) ) )
1110nrex 2524 . . . . . . . . . 10  |-  -.  E. m  e.  om  ( dom  (/)  =  suc  m  /\  x  e.  ( F `  ( (/) `  m
) ) )
1211biorfi 735 . . . . . . . . 9  |-  ( ( dom  (/)  =  (/)  /\  x  e.  A )  <->  ( ( dom  (/)  =  (/)  /\  x  e.  A )  \/  E. m  e.  om  ( dom  (/)  =  suc  m  /\  x  e.  ( F `  ( (/) `  m
) ) ) ) )
13 orcom 717 . . . . . . . . 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 2255 . . . . . . 7  |-  { x  |  x  e.  A }  =  { x  |  ( E. m  e.  om  ( dom  (/)  =  suc  m  /\  x  e.  ( F `  ( (/) `  m ) ) )  \/  ( dom  (/)  =  (/)  /\  x  e.  A ) ) }
16 abid2 2260 . . . . . . 7  |-  { x  |  x  e.  A }  =  A
1715, 16eqtr3i 2162 . . . . . 6  |-  { x  |  ( E. m  e.  om  ( dom  (/)  =  suc  m  /\  x  e.  ( F `  ( (/) `  m ) ) )  \/  ( dom  (/)  =  (/)  /\  x  e.  A ) ) }  =  A
18 elex 2697 . . . . . 6  |-  ( A  e.  V  ->  A  e.  _V )
1917, 18eqeltrid 2226 . . . . 5  |-  ( A  e.  V  ->  { x  |  ( E. m  e.  om  ( dom  (/)  =  suc  m  /\  x  e.  ( F `  ( (/) `  m ) ) )  \/  ( dom  (/)  =  (/)  /\  x  e.  A ) ) }  e.  _V )
20 0ex 4055 . . . . . . 7  |-  (/)  e.  _V
21 dmeq 4739 . . . . . . . . . . . . 13  |-  ( g  =  (/)  ->  dom  g  =  dom  (/) )
2221eqeq1d 2148 . . . . . . . . . . . 12  |-  ( g  =  (/)  ->  ( dom  g  =  suc  m  <->  dom  (/)  =  suc  m ) )
23 fveq1 5420 . . . . . . . . . . . . . 14  |-  ( g  =  (/)  ->  ( g `
 m )  =  ( (/) `  m ) )
2423fveq2d 5425 . . . . . . . . . . . . 13  |-  ( g  =  (/)  ->  ( F `
 ( g `  m ) )  =  ( F `  ( (/) `  m ) ) )
2524eleq2d 2209 . . . . . . . . . . . 12  |-  ( g  =  (/)  ->  ( x  e.  ( F `  ( g `  m
) )  <->  x  e.  ( F `  ( (/) `  m ) ) ) )
2622, 25anbi12d 464 . . . . . . . . . . 11  |-  ( g  =  (/)  ->  ( ( dom  g  =  suc  m  /\  x  e.  ( F `  ( g `
 m ) ) )  <->  ( dom  (/)  =  suc  m  /\  x  e.  ( F `  ( (/) `  m ) ) ) ) )
2726rexbidv 2438 . . . . . . . . . 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 2148 . . . . . . . . . . 11  |-  ( g  =  (/)  ->  ( dom  g  =  (/)  <->  dom  (/)  =  (/) ) )
2928anbi1d 460 . . . . . . . . . 10  |-  ( g  =  (/)  ->  ( ( dom  g  =  (/)  /\  x  e.  A )  <-> 
( dom  (/)  =  (/)  /\  x  e.  A ) ) )
3027, 29orbi12d 782 . . . . . . . . 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 2257 . . . . . . . 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 2139 . . . . . . . 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 5497 . . . . . . 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 420 . . . . . 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 2188 . . . . 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 2216 . . 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 6288 . . . . . 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 5422 . . . . 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 4508 . . . . . 6  |-  (/)  e.  om
41 fvres 5445 . . . . . 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 5 . . . . 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 2160 . . . 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 2139 . . . . 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 6220 . . . 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 2184 . . 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 2172 1  |-  ( A  e.  V  ->  (frec ( F ,  A ) `
 (/) )  =  A )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 103    \/ wo 697    = wceq 1331    e. wcel 1480   {cab 2125    =/= wne 2308   E.wrex 2417   _Vcvv 2686   (/)c0 3363    |-> cmpt 3989   suc csuc 4287   omcom 4504   dom cdm 4539    |` cres 4541   ` cfv 5123  recscrecs 6201  freccfrec 6287
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-nul 4054  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-ral 2421  df-rex 2422  df-rab 2425  df-v 2688  df-sbc 2910  df-csb 3004  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-nul 3364  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-int 3772  df-iun 3815  df-br 3930  df-opab 3990  df-mpt 3991  df-tr 4027  df-id 4215  df-iord 4288  df-on 4290  df-suc 4293  df-iom 4505  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-res 4551  df-iota 5088  df-fun 5125  df-fn 5126  df-fv 5131  df-recs 6202  df-frec 6288
This theorem is referenced by:  frecrdg  6305  frec2uz0d  10186  frec2uzrdg  10196  frecuzrdg0  10200  frecuzrdgg  10203  frecuzrdg0t  10209  seq3val  10245  seqvalcd  10246
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