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Theorem frecsuc 5991
Description: The successor value resulting from finite recursive definition generation. (Contributed by Jim Kingdon, 15-Aug-2019.)
Assertion
Ref Expression
frecsuc  |-  ( ( A. z ( F `
 z )  e. 
_V  /\  A  e.  V  /\  B  e.  om )  ->  (frec ( F ,  A ) `  suc  B )  =  ( F `  (frec ( F ,  A ) `
 B ) ) )
Distinct variable groups:    z, A    z, B    z, F
Allowed substitution hint:    V( 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 suceq 4139 . . . . . . . . . 10  |-  ( n  =  m  ->  suc  n  =  suc  m )
21eqeq2d 2051 . . . . . . . . 9  |-  ( n  =  m  ->  ( dom  f  =  suc  n 
<->  dom  f  =  suc  m ) )
3 fveq2 5178 . . . . . . . . . . 11  |-  ( n  =  m  ->  (
f `  n )  =  ( f `  m ) )
43fveq2d 5182 . . . . . . . . . 10  |-  ( n  =  m  ->  ( F `  ( f `  n ) )  =  ( F `  (
f `  m )
) )
54eleq2d 2107 . . . . . . . . 9  |-  ( n  =  m  ->  (
x  e.  ( F `
 ( f `  n ) )  <->  x  e.  ( F `  ( f `
 m ) ) ) )
62, 5anbi12d 442 . . . . . . . 8  |-  ( n  =  m  ->  (
( dom  f  =  suc  n  /\  x  e.  ( F `  (
f `  n )
) )  <->  ( dom  f  =  suc  m  /\  x  e.  ( F `  ( f `  m
) ) ) ) )
76cbvrexv 2534 . . . . . . 7  |-  ( E. n  e.  om  ( dom  f  =  suc  n  /\  x  e.  ( F `  ( f `
 n ) ) )  <->  E. m  e.  om  ( dom  f  =  suc  m  /\  x  e.  ( F `  ( f `
 m ) ) ) )
87orbi1i 680 . . . . . 6  |-  ( ( E. n  e.  om  ( dom  f  =  suc  n  /\  x  e.  ( F `  ( f `
 n ) ) )  \/  ( dom  f  =  (/)  /\  x  e.  A ) )  <->  ( E. m  e.  om  ( dom  f  =  suc  m  /\  x  e.  ( F `  ( f `
 m ) ) )  \/  ( dom  f  =  (/)  /\  x  e.  A ) ) )
98abbii 2153 . . . . 5  |-  { x  |  ( E. n  e.  om  ( dom  f  =  suc  n  /\  x  e.  ( F `  (
f `  n )
) )  \/  ( dom  f  =  (/)  /\  x  e.  A ) ) }  =  { x  |  ( E. m  e. 
om  ( dom  f  =  suc  m  /\  x  e.  ( F `  (
f `  m )
) )  \/  ( dom  f  =  (/)  /\  x  e.  A ) ) }
10 eleq1 2100 . . . . . . . . 9  |-  ( x  =  y  ->  (
x  e.  ( F `
 ( f `  m ) )  <->  y  e.  ( F `  ( f `
 m ) ) ) )
1110anbi2d 437 . . . . . . . 8  |-  ( x  =  y  ->  (
( dom  f  =  suc  m  /\  x  e.  ( F `  (
f `  m )
) )  <->  ( dom  f  =  suc  m  /\  y  e.  ( F `  ( f `  m
) ) ) ) )
1211rexbidv 2327 . . . . . . 7  |-  ( x  =  y  ->  ( E. m  e.  om  ( dom  f  =  suc  m  /\  x  e.  ( F `  ( f `
 m ) ) )  <->  E. m  e.  om  ( dom  f  =  suc  m  /\  y  e.  ( F `  ( f `
 m ) ) ) ) )
13 eleq1 2100 . . . . . . . 8  |-  ( x  =  y  ->  (
x  e.  A  <->  y  e.  A ) )
1413anbi2d 437 . . . . . . 7  |-  ( x  =  y  ->  (
( dom  f  =  (/) 
/\  x  e.  A
)  <->  ( dom  f  =  (/)  /\  y  e.  A ) ) )
1512, 14orbi12d 707 . . . . . 6  |-  ( x  =  y  ->  (
( E. m  e. 
om  ( dom  f  =  suc  m  /\  x  e.  ( F `  (
f `  m )
) )  \/  ( dom  f  =  (/)  /\  x  e.  A ) )  <->  ( E. m  e.  om  ( dom  f  =  suc  m  /\  y  e.  ( F `  ( f `
 m ) ) )  \/  ( dom  f  =  (/)  /\  y  e.  A ) ) ) )
1615cbvabv 2161 . . . . 5  |-  { x  |  ( E. m  e.  om  ( dom  f  =  suc  m  /\  x  e.  ( F `  (
f `  m )
) )  \/  ( dom  f  =  (/)  /\  x  e.  A ) ) }  =  { y  |  ( E. m  e. 
om  ( dom  f  =  suc  m  /\  y  e.  ( F `  (
f `  m )
) )  \/  ( dom  f  =  (/)  /\  y  e.  A ) ) }
179, 16eqtri 2060 . . . 4  |-  { x  |  ( E. n  e.  om  ( dom  f  =  suc  n  /\  x  e.  ( F `  (
f `  n )
) )  \/  ( dom  f  =  (/)  /\  x  e.  A ) ) }  =  { y  |  ( E. m  e. 
om  ( dom  f  =  suc  m  /\  y  e.  ( F `  (
f `  m )
) )  \/  ( dom  f  =  (/)  /\  y  e.  A ) ) }
1817mpteq2i 3844 . . 3  |-  ( f  e.  _V  |->  { x  |  ( E. n  e.  om  ( dom  f  =  suc  n  /\  x  e.  ( F `  (
f `  n )
) )  \/  ( dom  f  =  (/)  /\  x  e.  A ) ) } )  =  ( f  e.  _V  |->  { y  |  ( E. m  e.  om  ( dom  f  =  suc  m  /\  y  e.  ( F `  (
f `  m )
) )  \/  ( dom  f  =  (/)  /\  y  e.  A ) ) } )
19 dmeq 4535 . . . . . . . . 9  |-  ( f  =  g  ->  dom  f  =  dom  g )
2019eqeq1d 2048 . . . . . . . 8  |-  ( f  =  g  ->  ( dom  f  =  suc  m 
<->  dom  g  =  suc  m ) )
21 fveq1 5177 . . . . . . . . . 10  |-  ( f  =  g  ->  (
f `  m )  =  ( g `  m ) )
2221fveq2d 5182 . . . . . . . . 9  |-  ( f  =  g  ->  ( F `  ( f `  m ) )  =  ( F `  (
g `  m )
) )
2322eleq2d 2107 . . . . . . . 8  |-  ( f  =  g  ->  (
y  e.  ( F `
 ( f `  m ) )  <->  y  e.  ( F `  ( g `
 m ) ) ) )
2420, 23anbi12d 442 . . . . . . 7  |-  ( f  =  g  ->  (
( dom  f  =  suc  m  /\  y  e.  ( F `  (
f `  m )
) )  <->  ( dom  g  =  suc  m  /\  y  e.  ( F `  ( g `  m
) ) ) ) )
2524rexbidv 2327 . . . . . 6  |-  ( f  =  g  ->  ( E. m  e.  om  ( dom  f  =  suc  m  /\  y  e.  ( F `  ( f `
 m ) ) )  <->  E. m  e.  om  ( dom  g  =  suc  m  /\  y  e.  ( F `  ( g `
 m ) ) ) ) )
2619eqeq1d 2048 . . . . . . 7  |-  ( f  =  g  ->  ( dom  f  =  (/)  <->  dom  g  =  (/) ) )
2726anbi1d 438 . . . . . 6  |-  ( f  =  g  ->  (
( dom  f  =  (/) 
/\  y  e.  A
)  <->  ( dom  g  =  (/)  /\  y  e.  A ) ) )
2825, 27orbi12d 707 . . . . 5  |-  ( f  =  g  ->  (
( E. m  e. 
om  ( dom  f  =  suc  m  /\  y  e.  ( F `  (
f `  m )
) )  \/  ( dom  f  =  (/)  /\  y  e.  A ) )  <->  ( E. m  e.  om  ( dom  g  =  suc  m  /\  y  e.  ( F `  ( g `
 m ) ) )  \/  ( dom  g  =  (/)  /\  y  e.  A ) ) ) )
2928abbidv 2155 . . . 4  |-  ( f  =  g  ->  { y  |  ( E. m  e.  om  ( dom  f  =  suc  m  /\  y  e.  ( F `  (
f `  m )
) )  \/  ( dom  f  =  (/)  /\  y  e.  A ) ) }  =  { y  |  ( E. m  e. 
om  ( dom  g  =  suc  m  /\  y  e.  ( F `  (
g `  m )
) )  \/  ( dom  g  =  (/)  /\  y  e.  A ) ) } )
3029cbvmptv 3852 . . 3  |-  ( f  e.  _V  |->  { y  |  ( E. m  e.  om  ( dom  f  =  suc  m  /\  y  e.  ( F `  (
f `  m )
) )  \/  ( dom  f  =  (/)  /\  y  e.  A ) ) } )  =  ( g  e.  _V  |->  { y  |  ( E. m  e.  om  ( dom  g  =  suc  m  /\  y  e.  ( F `  (
g `  m )
) )  \/  ( dom  g  =  (/)  /\  y  e.  A ) ) } )
3118, 30eqtri 2060 . 2  |-  ( f  e.  _V  |->  { x  |  ( E. n  e.  om  ( dom  f  =  suc  n  /\  x  e.  ( F `  (
f `  n )
) )  \/  ( dom  f  =  (/)  /\  x  e.  A ) ) } )  =  ( g  e.  _V  |->  { y  |  ( E. m  e.  om  ( dom  g  =  suc  m  /\  y  e.  ( F `  (
g `  m )
) )  \/  ( dom  g  =  (/)  /\  y  e.  A ) ) } )
3231frecsuclem3 5990 1  |-  ( ( A. z ( F `
 z )  e. 
_V  /\  A  e.  V  /\  B  e.  om )  ->  (frec ( F ,  A ) `  suc  B )  =  ( F `  (frec ( F ,  A ) `
 B ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 97    \/ wo 629    /\ w3a 885   A.wal 1241    = wceq 1243    e. wcel 1393   {cab 2026   E.wrex 2307   _Vcvv 2557   (/)c0 3224    |-> cmpt 3818   suc csuc 4102   omcom 4313   dom cdm 4345   ` cfv 4902  freccfrec 5977
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-coll 3872  ax-sep 3875  ax-nul 3883  ax-pow 3927  ax-pr 3944  ax-un 4170  ax-setind 4262  ax-iinf 4311
This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-fal 1249  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ne 2206  df-ral 2311  df-rex 2312  df-reu 2313  df-rab 2315  df-v 2559  df-sbc 2765  df-csb 2853  df-dif 2920  df-un 2922  df-in 2924  df-ss 2931  df-nul 3225  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-int 3616  df-iun 3659  df-br 3765  df-opab 3819  df-mpt 3820  df-tr 3855  df-id 4030  df-iord 4103  df-on 4105  df-suc 4108  df-iom 4314  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-res 4357  df-ima 4358  df-iota 4867  df-fun 4904  df-fn 4905  df-f 4906  df-f1 4907  df-fo 4908  df-f1o 4909  df-fv 4910  df-recs 5920  df-frec 5978
This theorem is referenced by:  frecrdg  5992  freccl  5993  frec2uzzd  9184  frec2uzsucd  9185  frec2uzrdg  9193  frecuzrdgsuc  9199
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