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Theorem frec0g 6262
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 4723 . . . . . . . . . 10  |-  dom  (/)  =  (/)
21biantrur 301 . . . . . . . . 9  |-  ( x  e.  A  <->  ( dom  (/)  =  (/)  /\  x  e.  A ) )
3 vex 2663 . . . . . . . . . . . . . . . 16  |-  m  e. 
_V
4 nsuceq0g 4310 . . . . . . . . . . . . . . . 16  |-  ( m  e.  _V  ->  suc  m  =/=  (/) )
53, 4ax-mp 5 . . . . . . . . . . . . . . 15  |-  suc  m  =/=  (/)
65nesymi 2331 . . . . . . . . . . . . . 14  |-  -.  (/)  =  suc  m
71eqeq1i 2125 . . . . . . . . . . . . . 14  |-  ( dom  (/)  =  suc  m  <->  (/)  =  suc  m )
86, 7mtbir 645 . . . . . . . . . . . . 13  |-  -.  dom  (/)  =  suc  m
98intnanr 900 . . . . . . . . . . . 12  |-  -.  ( dom  (/)  =  suc  m  /\  x  e.  ( F `  ( (/) `  m
) ) )
109a1i 9 . . . . . . . . . . 11  |-  ( m  e.  om  ->  -.  ( dom  (/)  =  suc  m  /\  x  e.  ( F `  ( (/) `  m
) ) ) )
1110nrex 2501 . . . . . . . . . 10  |-  -.  E. m  e.  om  ( dom  (/)  =  suc  m  /\  x  e.  ( F `  ( (/) `  m
) ) )
1211biorfi 720 . . . . . . . . 9  |-  ( ( dom  (/)  =  (/)  /\  x  e.  A )  <->  ( ( dom  (/)  =  (/)  /\  x  e.  A )  \/  E. m  e.  om  ( dom  (/)  =  suc  m  /\  x  e.  ( F `  ( (/) `  m
) ) ) ) )
13 orcom 702 . . . . . . . . 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 2233 . . . . . . 7  |-  { x  |  x  e.  A }  =  { x  |  ( E. m  e.  om  ( dom  (/)  =  suc  m  /\  x  e.  ( F `  ( (/) `  m ) ) )  \/  ( dom  (/)  =  (/)  /\  x  e.  A ) ) }
16 abid2 2238 . . . . . . 7  |-  { x  |  x  e.  A }  =  A
1715, 16eqtr3i 2140 . . . . . 6  |-  { x  |  ( E. m  e.  om  ( dom  (/)  =  suc  m  /\  x  e.  ( F `  ( (/) `  m ) ) )  \/  ( dom  (/)  =  (/)  /\  x  e.  A ) ) }  =  A
18 elex 2671 . . . . . 6  |-  ( A  e.  V  ->  A  e.  _V )
1917, 18eqeltrid 2204 . . . . 5  |-  ( A  e.  V  ->  { x  |  ( E. m  e.  om  ( dom  (/)  =  suc  m  /\  x  e.  ( F `  ( (/) `  m ) ) )  \/  ( dom  (/)  =  (/)  /\  x  e.  A ) ) }  e.  _V )
20 0ex 4025 . . . . . . 7  |-  (/)  e.  _V
21 dmeq 4709 . . . . . . . . . . . . 13  |-  ( g  =  (/)  ->  dom  g  =  dom  (/) )
2221eqeq1d 2126 . . . . . . . . . . . 12  |-  ( g  =  (/)  ->  ( dom  g  =  suc  m  <->  dom  (/)  =  suc  m ) )
23 fveq1 5388 . . . . . . . . . . . . . 14  |-  ( g  =  (/)  ->  ( g `
 m )  =  ( (/) `  m ) )
2423fveq2d 5393 . . . . . . . . . . . . 13  |-  ( g  =  (/)  ->  ( F `
 ( g `  m ) )  =  ( F `  ( (/) `  m ) ) )
2524eleq2d 2187 . . . . . . . . . . . 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 2415 . . . . . . . . . 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 2126 . . . . . . . . . . 11  |-  ( g  =  (/)  ->  ( dom  g  =  (/)  <->  dom  (/)  =  (/) ) )
2928anbi1d 460 . . . . . . . . . 10  |-  ( g  =  (/)  ->  ( ( dom  g  =  (/)  /\  x  e.  A )  <-> 
( dom  (/)  =  (/)  /\  x  e.  A ) ) )
3027, 29orbi12d 767 . . . . . . . . 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 2235 . . . . . . . 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 2117 . . . . . . . 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 5465 . . . . . . 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 2166 . . . . 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 2194 . . 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 6256 . . . . . 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 5390 . . . . 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 4478 . . . . . 6  |-  (/)  e.  om
41 fvres 5413 . . . . . 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 2138 . . . 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 2117 . . . . 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 6188 . . . 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 2162 . . 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 2150 1  |-  ( A  e.  V  ->  (frec ( F ,  A ) `
 (/) )  =  A )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 103    \/ wo 682    = wceq 1316    e. wcel 1465   {cab 2103    =/= wne 2285   E.wrex 2394   _Vcvv 2660   (/)c0 3333    |-> cmpt 3959   suc csuc 4257   omcom 4474   dom cdm 4509    |` cres 4511   ` cfv 5093  recscrecs 6169  freccfrec 6255
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 588  ax-in2 589  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-13 1476  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016  ax-nul 4024  ax-pow 4068  ax-pr 4101  ax-un 4325  ax-setind 4422
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-fal 1322  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ne 2286  df-ral 2398  df-rex 2399  df-rab 2402  df-v 2662  df-sbc 2883  df-csb 2976  df-dif 3043  df-un 3045  df-in 3047  df-ss 3054  df-nul 3334  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-uni 3707  df-int 3742  df-iun 3785  df-br 3900  df-opab 3960  df-mpt 3961  df-tr 3997  df-id 4185  df-iord 4258  df-on 4260  df-suc 4263  df-iom 4475  df-xp 4515  df-rel 4516  df-cnv 4517  df-co 4518  df-dm 4519  df-res 4521  df-iota 5058  df-fun 5095  df-fn 5096  df-fv 5101  df-recs 6170  df-frec 6256
This theorem is referenced by:  frecrdg  6273  frec2uz0d  10140  frec2uzrdg  10150  frecuzrdg0  10154  frecuzrdgg  10157  frecuzrdg0t  10163  seq3val  10199  seqvalcd  10200
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