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Theorem frecsuc 6436
Description: The successor value resulting from finite recursive definition generation. (Contributed by Jim Kingdon, 31-Mar-2022.)
Assertion
Ref Expression
frecsuc  |-  ( ( A. z  e.  S  ( F `  z )  e.  S  /\  A  e.  S  /\  B  e. 
om )  ->  (frec ( F ,  A ) `
 suc  B )  =  ( F `  (frec ( F ,  A
) `  B )
) )
Distinct variable groups:    z, F    z, S
Allowed substitution hints:    A( z)    B( z)

Proof of Theorem frecsuc
Dummy variables  f  g  m  x  y  n are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 dmeq 4848 . . . . . . . . 9  |-  ( f  =  g  ->  dom  f  =  dom  g )
21eqeq1d 2198 . . . . . . . 8  |-  ( f  =  g  ->  ( dom  f  =  suc  n 
<->  dom  g  =  suc  n ) )
3 fveq1 5536 . . . . . . . . . 10  |-  ( f  =  g  ->  (
f `  n )  =  ( g `  n ) )
43fveq2d 5541 . . . . . . . . 9  |-  ( f  =  g  ->  ( F `  ( f `  n ) )  =  ( F `  (
g `  n )
) )
54eleq2d 2259 . . . . . . . 8  |-  ( f  =  g  ->  (
y  e.  ( F `
 ( f `  n ) )  <->  y  e.  ( F `  ( g `
 n ) ) ) )
62, 5anbi12d 473 . . . . . . 7  |-  ( f  =  g  ->  (
( dom  f  =  suc  n  /\  y  e.  ( F `  (
f `  n )
) )  <->  ( dom  g  =  suc  n  /\  y  e.  ( F `  ( g `  n
) ) ) ) )
76rexbidv 2491 . . . . . 6  |-  ( f  =  g  ->  ( E. n  e.  om  ( dom  f  =  suc  n  /\  y  e.  ( F `  ( f `
 n ) ) )  <->  E. n  e.  om  ( dom  g  =  suc  n  /\  y  e.  ( F `  ( g `
 n ) ) ) ) )
81eqeq1d 2198 . . . . . . 7  |-  ( f  =  g  ->  ( dom  f  =  (/)  <->  dom  g  =  (/) ) )
98anbi1d 465 . . . . . 6  |-  ( f  =  g  ->  (
( dom  f  =  (/) 
/\  y  e.  A
)  <->  ( dom  g  =  (/)  /\  y  e.  A ) ) )
107, 9orbi12d 794 . . . . 5  |-  ( f  =  g  ->  (
( E. n  e. 
om  ( dom  f  =  suc  n  /\  y  e.  ( F `  (
f `  n )
) )  \/  ( dom  f  =  (/)  /\  y  e.  A ) )  <->  ( E. n  e.  om  ( dom  g  =  suc  n  /\  y  e.  ( F `  ( g `
 n ) ) )  \/  ( dom  g  =  (/)  /\  y  e.  A ) ) ) )
1110abbidv 2307 . . . 4  |-  ( f  =  g  ->  { y  |  ( E. n  e.  om  ( dom  f  =  suc  n  /\  y  e.  ( F `  (
f `  n )
) )  \/  ( dom  f  =  (/)  /\  y  e.  A ) ) }  =  { y  |  ( E. n  e. 
om  ( dom  g  =  suc  n  /\  y  e.  ( F `  (
g `  n )
) )  \/  ( dom  g  =  (/)  /\  y  e.  A ) ) } )
1211cbvmptv 4117 . . 3  |-  ( f  e.  _V  |->  { y  |  ( E. n  e.  om  ( dom  f  =  suc  n  /\  y  e.  ( F `  (
f `  n )
) )  \/  ( dom  f  =  (/)  /\  y  e.  A ) ) } )  =  ( g  e.  _V  |->  { y  |  ( E. n  e.  om  ( dom  g  =  suc  n  /\  y  e.  ( F `  (
g `  n )
) )  \/  ( dom  g  =  (/)  /\  y  e.  A ) ) } )
13 eleq1 2252 . . . . . . . 8  |-  ( y  =  x  ->  (
y  e.  ( F `
 ( g `  n ) )  <->  x  e.  ( F `  ( g `
 n ) ) ) )
1413anbi2d 464 . . . . . . 7  |-  ( y  =  x  ->  (
( dom  g  =  suc  n  /\  y  e.  ( F `  (
g `  n )
) )  <->  ( dom  g  =  suc  n  /\  x  e.  ( F `  ( g `  n
) ) ) ) )
1514rexbidv 2491 . . . . . 6  |-  ( y  =  x  ->  ( E. n  e.  om  ( dom  g  =  suc  n  /\  y  e.  ( F `  ( g `
 n ) ) )  <->  E. n  e.  om  ( dom  g  =  suc  n  /\  x  e.  ( F `  ( g `
 n ) ) ) ) )
16 eleq1 2252 . . . . . . 7  |-  ( y  =  x  ->  (
y  e.  A  <->  x  e.  A ) )
1716anbi2d 464 . . . . . 6  |-  ( y  =  x  ->  (
( dom  g  =  (/) 
/\  y  e.  A
)  <->  ( dom  g  =  (/)  /\  x  e.  A ) ) )
1815, 17orbi12d 794 . . . . 5  |-  ( y  =  x  ->  (
( E. n  e. 
om  ( dom  g  =  suc  n  /\  y  e.  ( F `  (
g `  n )
) )  \/  ( dom  g  =  (/)  /\  y  e.  A ) )  <->  ( E. n  e.  om  ( dom  g  =  suc  n  /\  x  e.  ( F `  ( g `
 n ) ) )  \/  ( dom  g  =  (/)  /\  x  e.  A ) ) ) )
1918cbvabv 2314 . . . 4  |-  { y  |  ( E. n  e.  om  ( dom  g  =  suc  n  /\  y  e.  ( F `  (
g `  n )
) )  \/  ( dom  g  =  (/)  /\  y  e.  A ) ) }  =  { x  |  ( E. n  e. 
om  ( dom  g  =  suc  n  /\  x  e.  ( F `  (
g `  n )
) )  \/  ( dom  g  =  (/)  /\  x  e.  A ) ) }
2019mpteq2i 4108 . . 3  |-  ( g  e.  _V  |->  { y  |  ( E. n  e.  om  ( dom  g  =  suc  n  /\  y  e.  ( F `  (
g `  n )
) )  \/  ( dom  g  =  (/)  /\  y  e.  A ) ) } )  =  ( g  e.  _V  |->  { x  |  ( E. n  e.  om  ( dom  g  =  suc  n  /\  x  e.  ( F `  (
g `  n )
) )  \/  ( dom  g  =  (/)  /\  x  e.  A ) ) } )
21 suceq 4423 . . . . . . . . 9  |-  ( n  =  m  ->  suc  n  =  suc  m )
2221eqeq2d 2201 . . . . . . . 8  |-  ( n  =  m  ->  ( dom  g  =  suc  n 
<->  dom  g  =  suc  m ) )
23 fveq2 5537 . . . . . . . . . 10  |-  ( n  =  m  ->  (
g `  n )  =  ( g `  m ) )
2423fveq2d 5541 . . . . . . . . 9  |-  ( n  =  m  ->  ( F `  ( g `  n ) )  =  ( F `  (
g `  m )
) )
2524eleq2d 2259 . . . . . . . 8  |-  ( n  =  m  ->  (
x  e.  ( F `
 ( g `  n ) )  <->  x  e.  ( F `  ( g `
 m ) ) ) )
2622, 25anbi12d 473 . . . . . . 7  |-  ( n  =  m  ->  (
( dom  g  =  suc  n  /\  x  e.  ( F `  (
g `  n )
) )  <->  ( dom  g  =  suc  m  /\  x  e.  ( F `  ( g `  m
) ) ) ) )
2726cbvrexv 2719 . . . . . 6  |-  ( E. n  e.  om  ( dom  g  =  suc  n  /\  x  e.  ( F `  ( g `
 n ) ) )  <->  E. m  e.  om  ( dom  g  =  suc  m  /\  x  e.  ( F `  ( g `
 m ) ) ) )
2827orbi1i 764 . . . . 5  |-  ( ( E. n  e.  om  ( dom  g  =  suc  n  /\  x  e.  ( F `  ( g `
 n ) ) )  \/  ( dom  g  =  (/)  /\  x  e.  A ) )  <->  ( E. m  e.  om  ( dom  g  =  suc  m  /\  x  e.  ( F `  ( g `
 m ) ) )  \/  ( dom  g  =  (/)  /\  x  e.  A ) ) )
2928abbii 2305 . . . 4  |-  { x  |  ( E. n  e.  om  ( dom  g  =  suc  n  /\  x  e.  ( F `  (
g `  n )
) )  \/  ( dom  g  =  (/)  /\  x  e.  A ) ) }  =  { x  |  ( E. m  e. 
om  ( dom  g  =  suc  m  /\  x  e.  ( F `  (
g `  m )
) )  \/  ( dom  g  =  (/)  /\  x  e.  A ) ) }
3029mpteq2i 4108 . . 3  |-  ( g  e.  _V  |->  { x  |  ( E. n  e.  om  ( dom  g  =  suc  n  /\  x  e.  ( F `  (
g `  n )
) )  \/  ( 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 ) ) } )
3112, 20, 303eqtri 2214 . 2  |-  ( f  e.  _V  |->  { y  |  ( E. n  e.  om  ( dom  f  =  suc  n  /\  y  e.  ( F `  (
f `  n )
) )  \/  ( dom  f  =  (/)  /\  y  e.  A ) ) } )  =  ( g  e.  _V  |->  { x  |  ( E. m  e.  om  ( dom  g  =  suc  m  /\  x  e.  ( F `  (
g `  m )
) )  \/  ( dom  g  =  (/)  /\  x  e.  A ) ) } )
3231frecsuclem 6435 1  |-  ( ( A. z  e.  S  ( F `  z )  e.  S  /\  A  e.  S  /\  B  e. 
om )  ->  (frec ( F ,  A ) `
 suc  B )  =  ( F `  (frec ( F ,  A
) `  B )
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    \/ wo 709    /\ w3a 980    = wceq 1364    e. wcel 2160   {cab 2175   A.wral 2468   E.wrex 2469   _Vcvv 2752   (/)c0 3437    |-> cmpt 4082   suc csuc 4386   omcom 4610   dom cdm 4647   ` cfv 5238  freccfrec 6419
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-coll 4136  ax-sep 4139  ax-nul 4147  ax-pow 4195  ax-pr 4230  ax-un 4454  ax-setind 4557  ax-iinf 4608
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-ral 2473  df-rex 2474  df-reu 2475  df-rab 2477  df-v 2754  df-sbc 2978  df-csb 3073  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-nul 3438  df-pw 3595  df-sn 3616  df-pr 3617  df-op 3619  df-uni 3828  df-int 3863  df-iun 3906  df-br 4022  df-opab 4083  df-mpt 4084  df-tr 4120  df-id 4314  df-iord 4387  df-on 4389  df-ilim 4390  df-suc 4392  df-iom 4611  df-xp 4653  df-rel 4654  df-cnv 4655  df-co 4656  df-dm 4657  df-rn 4658  df-res 4659  df-ima 4660  df-iota 5199  df-fun 5240  df-fn 5241  df-f 5242  df-f1 5243  df-fo 5244  df-f1o 5245  df-fv 5246  df-recs 6334  df-frec 6420
This theorem is referenced by:  frecrdg  6437  frec2uzsucd  10438  frec2uzrdg  10446  frecuzrdgsuc  10451  frecuzrdgg  10453  frecuzrdgsuctlem  10460  seq3val  10497  seqvalcd  10498
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