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Theorem frec0g 6014
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 4577 . . . . . . . . . 10  |-  dom  (/)  =  (/)
21biantrur 291 . . . . . . . . 9  |-  ( x  e.  A  <->  ( dom  (/)  =  (/)  /\  x  e.  A ) )
3 vex 2577 . . . . . . . . . . . . . . . 16  |-  m  e. 
_V
4 nsuceq0g 4183 . . . . . . . . . . . . . . . 16  |-  ( m  e.  _V  ->  suc  m  =/=  (/) )
53, 4ax-mp 7 . . . . . . . . . . . . . . 15  |-  suc  m  =/=  (/)
65nesymi 2266 . . . . . . . . . . . . . 14  |-  -.  (/)  =  suc  m
71eqeq1i 2063 . . . . . . . . . . . . . 14  |-  ( dom  (/)  =  suc  m  <->  (/)  =  suc  m )
86, 7mtbir 606 . . . . . . . . . . . . 13  |-  -.  dom  (/)  =  suc  m
98intnanr 850 . . . . . . . . . . . 12  |-  -.  ( dom  (/)  =  suc  m  /\  x  e.  ( F `  ( (/) `  m
) ) )
109a1i 9 . . . . . . . . . . 11  |-  ( m  e.  om  ->  -.  ( dom  (/)  =  suc  m  /\  x  e.  ( F `  ( (/) `  m
) ) ) )
1110nrex 2428 . . . . . . . . . 10  |-  -.  E. m  e.  om  ( dom  (/)  =  suc  m  /\  x  e.  ( F `  ( (/) `  m
) ) )
1211biorfi 675 . . . . . . . . 9  |-  ( ( dom  (/)  =  (/)  /\  x  e.  A )  <->  ( ( dom  (/)  =  (/)  /\  x  e.  A )  \/  E. m  e.  om  ( dom  (/)  =  suc  m  /\  x  e.  ( F `  ( (/) `  m
) ) ) ) )
13 orcom 657 . . . . . . . . 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 199 . . . . . . . 8  |-  ( x  e.  A  <->  ( E. m  e.  om  ( dom  (/)  =  suc  m  /\  x  e.  ( F `  ( (/) `  m
) ) )  \/  ( dom  (/)  =  (/)  /\  x  e.  A ) ) )
1514abbii 2169 . . . . . . 7  |-  { x  |  x  e.  A }  =  { x  |  ( E. m  e.  om  ( dom  (/)  =  suc  m  /\  x  e.  ( F `  ( (/) `  m ) ) )  \/  ( dom  (/)  =  (/)  /\  x  e.  A ) ) }
16 abid2 2174 . . . . . . 7  |-  { x  |  x  e.  A }  =  A
1715, 16eqtr3i 2078 . . . . . 6  |-  { x  |  ( E. m  e.  om  ( dom  (/)  =  suc  m  /\  x  e.  ( F `  ( (/) `  m ) ) )  \/  ( dom  (/)  =  (/)  /\  x  e.  A ) ) }  =  A
18 elex 2583 . . . . . 6  |-  ( A  e.  V  ->  A  e.  _V )
1917, 18syl5eqel 2140 . . . . 5  |-  ( A  e.  V  ->  { x  |  ( E. m  e.  om  ( dom  (/)  =  suc  m  /\  x  e.  ( F `  ( (/) `  m ) ) )  \/  ( dom  (/)  =  (/)  /\  x  e.  A ) ) }  e.  _V )
20 0ex 3912 . . . . . . 7  |-  (/)  e.  _V
21 dmeq 4563 . . . . . . . . . . . . 13  |-  ( g  =  (/)  ->  dom  g  =  dom  (/) )
2221eqeq1d 2064 . . . . . . . . . . . 12  |-  ( g  =  (/)  ->  ( dom  g  =  suc  m  <->  dom  (/)  =  suc  m ) )
23 fveq1 5205 . . . . . . . . . . . . . 14  |-  ( g  =  (/)  ->  ( g `
 m )  =  ( (/) `  m ) )
2423fveq2d 5210 . . . . . . . . . . . . 13  |-  ( g  =  (/)  ->  ( F `
 ( g `  m ) )  =  ( F `  ( (/) `  m ) ) )
2524eleq2d 2123 . . . . . . . . . . . 12  |-  ( g  =  (/)  ->  ( x  e.  ( F `  ( g `  m
) )  <->  x  e.  ( F `  ( (/) `  m ) ) ) )
2622, 25anbi12d 450 . . . . . . . . . . 11  |-  ( g  =  (/)  ->  ( ( dom  g  =  suc  m  /\  x  e.  ( F `  ( g `
 m ) ) )  <->  ( dom  (/)  =  suc  m  /\  x  e.  ( F `  ( (/) `  m ) ) ) ) )
2726rexbidv 2344 . . . . . . . . . 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 2064 . . . . . . . . . . 11  |-  ( g  =  (/)  ->  ( dom  g  =  (/)  <->  dom  (/)  =  (/) ) )
2928anbi1d 446 . . . . . . . . . 10  |-  ( g  =  (/)  ->  ( ( dom  g  =  (/)  /\  x  e.  A )  <-> 
( dom  (/)  =  (/)  /\  x  e.  A ) ) )
3027, 29orbi12d 717 . . . . . . . . 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 2171 . . . . . . . 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 2056 . . . . . . . 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 5276 . . . . . . 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 408 . . . . . 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 2104 . . . . 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 2130 . . 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 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 )
3938fveq1i 5207 . . . . 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 4345 . . . . . 6  |-  (/)  e.  om
41 fvres 5226 . . . . . 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 2076 . . . 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 2056 . . . . 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 5968 . . . 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 2100 . . 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 2088 1  |-  ( A  e.  V  ->  (frec ( F ,  A ) `
 (/) )  =  A )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 101    \/ wo 639    = wceq 1259    e. wcel 1409   {cab 2042    =/= wne 2220   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-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-sep 3903  ax-nul 3911  ax-pow 3955  ax-pr 3972  ax-un 4198  ax-setind 4290
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-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-res 4385  df-iota 4895  df-fun 4932  df-fn 4933  df-fv 4938  df-recs 5951  df-frec 6009
This theorem is referenced by:  frecrdg  6023  freccl  6024  frec2uz0d  9349  frec2uzrdg  9359  frecuzrdg0  9364
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