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Theorem freccllem 6123
Description: Lemma for freccl 6124. Just giving a name to a common expression to simplify the proof. (Contributed by Jim Kingdon, 27-Mar-2022.)
Hypotheses
Ref Expression
freccl.a  |-  ( ph  ->  A  e.  S )
freccl.cl  |-  ( (
ph  /\  z  e.  S )  ->  ( F `  z )  e.  S )
freccl.b  |-  ( ph  ->  B  e.  om )
freccllem.g  |-  G  = recs ( ( g  e. 
_V  |->  { x  |  ( E. m  e. 
om  ( dom  g  =  suc  m  /\  x  e.  ( F `  (
g `  m )
) )  \/  ( dom  g  =  (/)  /\  x  e.  A ) ) } ) )
Assertion
Ref Expression
freccllem  |-  ( ph  ->  (frec ( F ,  A ) `  B
)  e.  S )
Distinct variable groups:    A, g, m, x    z, A, m, x    x, B    g, F, m, x    z, F    S, m, x, z    ph, m, x, z
Allowed substitution hints:    ph( g)    B( z,
g, m)    S( g)    G( x, z, g, m)

Proof of Theorem freccllem
Dummy variables  f  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-frec 6112 . . . 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 ) ) } ) )  |`  om )
2 freccllem.g . . . . 5  |-  G  = recs ( ( g  e. 
_V  |->  { x  |  ( E. m  e. 
om  ( dom  g  =  suc  m  /\  x  e.  ( F `  (
g `  m )
) )  \/  ( dom  g  =  (/)  /\  x  e.  A ) ) } ) )
32reseq1i 4679 . . . 4  |-  ( G  |`  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 )
41, 3eqtr4i 2108 . . 3  |- frec ( F ,  A )  =  ( G  |`  om )
54fveq1i 5271 . 2  |-  (frec ( F ,  A ) `
 B )  =  ( ( G  |`  om ) `  B )
6 freccl.b . . . 4  |-  ( ph  ->  B  e.  om )
7 fvres 5294 . . . 4  |-  ( B  e.  om  ->  (
( G  |`  om ) `  B )  =  ( G `  B ) )
86, 7syl 14 . . 3  |-  ( ph  ->  ( ( G  |`  om ) `  B )  =  ( G `  B ) )
9 funmpt 5019 . . . . 5  |-  Fun  (
g  e.  _V  |->  { x  |  ( E. m  e.  om  ( dom  g  =  suc  m  /\  x  e.  ( F `  ( g `
 m ) ) )  \/  ( dom  g  =  (/)  /\  x  e.  A ) ) } )
109a1i 9 . . . 4  |-  ( ph  ->  Fun  ( g  e. 
_V  |->  { x  |  ( E. m  e. 
om  ( dom  g  =  suc  m  /\  x  e.  ( F `  (
g `  m )
) )  \/  ( dom  g  =  (/)  /\  x  e.  A ) ) } ) )
11 ordom 4396 . . . . 5  |-  Ord  om
1211a1i 9 . . . 4  |-  ( ph  ->  Ord  om )
13 vex 2618 . . . . . 6  |-  f  e. 
_V
14 simp2 942 . . . . . . 7  |-  ( (
ph  /\  y  e.  om 
/\  f : y --> S )  ->  y  e.  om )
15 simp3 943 . . . . . . 7  |-  ( (
ph  /\  y  e.  om 
/\  f : y --> S )  ->  f : y --> S )
16 freccl.cl . . . . . . . . 9  |-  ( (
ph  /\  z  e.  S )  ->  ( F `  z )  e.  S )
1716ralrimiva 2442 . . . . . . . 8  |-  ( ph  ->  A. z  e.  S  ( F `  z )  e.  S )
18173ad2ant1 962 . . . . . . 7  |-  ( (
ph  /\  y  e.  om 
/\  f : y --> S )  ->  A. z  e.  S  ( F `  z )  e.  S
)
19 freccl.a . . . . . . . 8  |-  ( ph  ->  A  e.  S )
20193ad2ant1 962 . . . . . . 7  |-  ( (
ph  /\  y  e.  om 
/\  f : y --> S )  ->  A  e.  S )
2114, 15, 18, 20frecabcl 6120 . . . . . 6  |-  ( (
ph  /\  y  e.  om 
/\  f : y --> S )  ->  { x  |  ( E. m  e.  om  ( dom  f  =  suc  m  /\  x  e.  ( F `  (
f `  m )
) )  \/  ( dom  f  =  (/)  /\  x  e.  A ) ) }  e.  S )
22 dmeq 4606 . . . . . . . . . . . 12  |-  ( g  =  f  ->  dom  g  =  dom  f )
2322eqeq1d 2093 . . . . . . . . . . 11  |-  ( g  =  f  ->  ( dom  g  =  suc  m 
<->  dom  f  =  suc  m ) )
24 fveq1 5269 . . . . . . . . . . . . 13  |-  ( g  =  f  ->  (
g `  m )  =  ( f `  m ) )
2524fveq2d 5274 . . . . . . . . . . . 12  |-  ( g  =  f  ->  ( F `  ( g `  m ) )  =  ( F `  (
f `  m )
) )
2625eleq2d 2154 . . . . . . . . . . 11  |-  ( g  =  f  ->  (
x  e.  ( F `
 ( g `  m ) )  <->  x  e.  ( F `  ( f `
 m ) ) ) )
2723, 26anbi12d 457 . . . . . . . . . 10  |-  ( g  =  f  ->  (
( dom  g  =  suc  m  /\  x  e.  ( F `  (
g `  m )
) )  <->  ( dom  f  =  suc  m  /\  x  e.  ( F `  ( f `  m
) ) ) ) )
2827rexbidv 2377 . . . . . . . . 9  |-  ( g  =  f  ->  ( E. m  e.  om  ( dom  g  =  suc  m  /\  x  e.  ( F `  ( g `
 m ) ) )  <->  E. m  e.  om  ( dom  f  =  suc  m  /\  x  e.  ( F `  ( f `
 m ) ) ) ) )
2922eqeq1d 2093 . . . . . . . . . 10  |-  ( g  =  f  ->  ( dom  g  =  (/)  <->  dom  f  =  (/) ) )
3029anbi1d 453 . . . . . . . . 9  |-  ( g  =  f  ->  (
( dom  g  =  (/) 
/\  x  e.  A
)  <->  ( dom  f  =  (/)  /\  x  e.  A ) ) )
3128, 30orbi12d 740 . . . . . . . 8  |-  ( g  =  f  ->  (
( E. m  e. 
om  ( dom  g  =  suc  m  /\  x  e.  ( F `  (
g `  m )
) )  \/  ( dom  g  =  (/)  /\  x  e.  A ) )  <->  ( E. m  e.  om  ( dom  f  =  suc  m  /\  x  e.  ( F `  ( f `
 m ) ) )  \/  ( dom  f  =  (/)  /\  x  e.  A ) ) ) )
3231abbidv 2202 . . . . . . 7  |-  ( g  =  f  ->  { 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  f  =  suc  m  /\  x  e.  ( F `  (
f `  m )
) )  \/  ( dom  f  =  (/)  /\  x  e.  A ) ) } )
33 eqid 2085 . . . . . . 7  |-  ( 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 ) ) } )
3432, 33fvmptg 5345 . . . . . 6  |-  ( ( f  e.  _V  /\  { x  |  ( E. m  e.  om  ( dom  f  =  suc  m  /\  x  e.  ( F `  ( f `
 m ) ) )  \/  ( dom  f  =  (/)  /\  x  e.  A ) ) }  e.  S )  -> 
( ( g  e. 
_V  |->  { x  |  ( E. m  e. 
om  ( dom  g  =  suc  m  /\  x  e.  ( F `  (
g `  m )
) )  \/  ( dom  g  =  (/)  /\  x  e.  A ) ) } ) `  f )  =  { x  |  ( E. m  e. 
om  ( dom  f  =  suc  m  /\  x  e.  ( F `  (
f `  m )
) )  \/  ( dom  f  =  (/)  /\  x  e.  A ) ) } )
3513, 21, 34sylancr 405 . . . . 5  |-  ( (
ph  /\  y  e.  om 
/\  f : y --> S )  ->  (
( g  e.  _V  |->  { x  |  ( E. m  e.  om  ( dom  g  =  suc  m  /\  x  e.  ( F `  ( g `
 m ) ) )  \/  ( dom  g  =  (/)  /\  x  e.  A ) ) } ) `  f )  =  { x  |  ( E. m  e. 
om  ( dom  f  =  suc  m  /\  x  e.  ( F `  (
f `  m )
) )  \/  ( dom  f  =  (/)  /\  x  e.  A ) ) } )
3635, 21eqeltrd 2161 . . . 4  |-  ( (
ph  /\  y  e.  om 
/\  f : y --> S )  ->  (
( g  e.  _V  |->  { x  |  ( E. m  e.  om  ( dom  g  =  suc  m  /\  x  e.  ( F `  ( g `
 m ) ) )  \/  ( dom  g  =  (/)  /\  x  e.  A ) ) } ) `  f )  e.  S )
37 limom 4403 . . . . . . 7  |-  Lim  om
38 limuni 4199 . . . . . . 7  |-  ( Lim 
om  ->  om  =  U. om )
3937, 38ax-mp 7 . . . . . 6  |-  om  =  U. om
4039eleq2i 2151 . . . . 5  |-  ( y  e.  om  <->  y  e.  U.
om )
41 peano2 4385 . . . . . 6  |-  ( y  e.  om  ->  suc  y  e.  om )
4241adantl 271 . . . . 5  |-  ( (
ph  /\  y  e.  om )  ->  suc  y  e. 
om )
4340, 42sylan2br 282 . . . 4  |-  ( (
ph  /\  y  e.  U.
om )  ->  suc  y  e.  om )
446, 39syl6eleq 2177 . . . 4  |-  ( ph  ->  B  e.  U. om )
452, 10, 12, 36, 43, 44tfrcl 6085 . . 3  |-  ( ph  ->  ( G `  B
)  e.  S )
468, 45eqeltrd 2161 . 2  |-  ( ph  ->  ( ( G  |`  om ) `  B )  e.  S )
475, 46syl5eqel 2171 1  |-  ( ph  ->  (frec ( F ,  A ) `  B
)  e.  S )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    \/ wo 662    /\ w3a 922    = wceq 1287    e. wcel 1436   {cab 2071   A.wral 2355   E.wrex 2356   _Vcvv 2615   (/)c0 3275   U.cuni 3638    |-> cmpt 3876   Ord word 4165   Lim wlim 4167   suc csuc 4168   omcom 4380   dom cdm 4413    |` cres 4415   Fun wfun 4977   -->wf 4979   ` cfv 4983  recscrecs 6025  freccfrec 6111
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-13 1447  ax-14 1448  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067  ax-coll 3931  ax-sep 3934  ax-nul 3942  ax-pow 3986  ax-pr 4012  ax-un 4236  ax-setind 4328  ax-iinf 4378
This theorem depends on definitions:  df-bi 115  df-3an 924  df-tru 1290  df-fal 1293  df-nf 1393  df-sb 1690  df-eu 1948  df-mo 1949  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-ne 2252  df-ral 2360  df-rex 2361  df-reu 2362  df-rab 2364  df-v 2617  df-sbc 2830  df-csb 2923  df-dif 2990  df-un 2992  df-in 2994  df-ss 3001  df-nul 3276  df-pw 3417  df-sn 3437  df-pr 3438  df-op 3440  df-uni 3639  df-int 3674  df-iun 3717  df-br 3823  df-opab 3877  df-mpt 3878  df-tr 3914  df-id 4096  df-iord 4169  df-on 4171  df-ilim 4172  df-suc 4174  df-iom 4381  df-xp 4419  df-rel 4420  df-cnv 4421  df-co 4422  df-dm 4423  df-rn 4424  df-res 4425  df-ima 4426  df-iota 4948  df-fun 4985  df-fn 4986  df-f 4987  df-f1 4988  df-fo 4989  df-f1o 4990  df-fv 4991  df-recs 6026  df-frec 6112
This theorem is referenced by:  freccl  6124
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