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Theorem freccllem 6402
Description: Lemma for freccl 6403. 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 6391 . . . 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 4903 . . . 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 2201 . . 3  |- frec ( F ,  A )  =  ( G  |`  om )
54fveq1i 5516 . 2  |-  (frec ( F ,  A ) `
 B )  =  ( ( G  |`  om ) `  B )
6 freccl.b . . . 4  |-  ( ph  ->  B  e.  om )
7 fvres 5539 . . . 4  |-  ( B  e.  om  ->  (
( G  |`  om ) `  B )  =  ( G `  B ) )
86, 7syl 14 . . 3  |-  ( ph  ->  ( ( G  |`  om ) `  B )  =  ( G `  B ) )
9 funmpt 5254 . . . . 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 4606 . . . . 5  |-  Ord  om
1211a1i 9 . . . 4  |-  ( ph  ->  Ord  om )
13 vex 2740 . . . . . 6  |-  f  e. 
_V
14 simp2 998 . . . . . . 7  |-  ( (
ph  /\  y  e.  om 
/\  f : y --> S )  ->  y  e.  om )
15 simp3 999 . . . . . . 7  |-  ( (
ph  /\  y  e.  om 
/\  f : y --> S )  ->  f : y --> S )
16 freccl.cl . . . . . . . . 9  |-  ( (
ph  /\  z  e.  S )  ->  ( F `  z )  e.  S )
1716ralrimiva 2550 . . . . . . . 8  |-  ( ph  ->  A. z  e.  S  ( F `  z )  e.  S )
18173ad2ant1 1018 . . . . . . 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 1018 . . . . . . 7  |-  ( (
ph  /\  y  e.  om 
/\  f : y --> S )  ->  A  e.  S )
2114, 15, 18, 20frecabcl 6399 . . . . . 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 4827 . . . . . . . . . . . 12  |-  ( g  =  f  ->  dom  g  =  dom  f )
2322eqeq1d 2186 . . . . . . . . . . 11  |-  ( g  =  f  ->  ( dom  g  =  suc  m 
<->  dom  f  =  suc  m ) )
24 fveq1 5514 . . . . . . . . . . . . 13  |-  ( g  =  f  ->  (
g `  m )  =  ( f `  m ) )
2524fveq2d 5519 . . . . . . . . . . . 12  |-  ( g  =  f  ->  ( F `  ( g `  m ) )  =  ( F `  (
f `  m )
) )
2625eleq2d 2247 . . . . . . . . . . 11  |-  ( g  =  f  ->  (
x  e.  ( F `
 ( g `  m ) )  <->  x  e.  ( F `  ( f `
 m ) ) ) )
2723, 26anbi12d 473 . . . . . . . . . 10  |-  ( g  =  f  ->  (
( dom  g  =  suc  m  /\  x  e.  ( F `  (
g `  m )
) )  <->  ( dom  f  =  suc  m  /\  x  e.  ( F `  ( f `  m
) ) ) ) )
2827rexbidv 2478 . . . . . . . . 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 2186 . . . . . . . . . 10  |-  ( g  =  f  ->  ( dom  g  =  (/)  <->  dom  f  =  (/) ) )
3029anbi1d 465 . . . . . . . . 9  |-  ( g  =  f  ->  (
( dom  g  =  (/) 
/\  x  e.  A
)  <->  ( dom  f  =  (/)  /\  x  e.  A ) ) )
3128, 30orbi12d 793 . . . . . . . 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 2295 . . . . . . 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 2177 . . . . . . 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 5592 . . . . . 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 414 . . . . 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 2254 . . . 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 4613 . . . . . . 7  |-  Lim  om
38 limuni 4396 . . . . . . 7  |-  ( Lim 
om  ->  om  =  U. om )
3937, 38ax-mp 5 . . . . . 6  |-  om  =  U. om
4039eleq2i 2244 . . . . 5  |-  ( y  e.  om  <->  y  e.  U.
om )
41 peano2 4594 . . . . . 6  |-  ( y  e.  om  ->  suc  y  e.  om )
4241adantl 277 . . . . 5  |-  ( (
ph  /\  y  e.  om )  ->  suc  y  e. 
om )
4340, 42sylan2br 288 . . . 4  |-  ( (
ph  /\  y  e.  U.
om )  ->  suc  y  e.  om )
446, 39eleqtrdi 2270 . . . 4  |-  ( ph  ->  B  e.  U. om )
452, 10, 12, 36, 43, 44tfrcl 6364 . . 3  |-  ( ph  ->  ( G `  B
)  e.  S )
468, 45eqeltrd 2254 . 2  |-  ( ph  ->  ( ( G  |`  om ) `  B )  e.  S )
475, 46eqeltrid 2264 1  |-  ( ph  ->  (frec ( F ,  A ) `  B
)  e.  S )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    \/ wo 708    /\ w3a 978    = wceq 1353    e. wcel 2148   {cab 2163   A.wral 2455   E.wrex 2456   _Vcvv 2737   (/)c0 3422   U.cuni 3809    |-> cmpt 4064   Ord word 4362   Lim wlim 4364   suc csuc 4365   omcom 4589   dom cdm 4626    |` cres 4628   Fun wfun 5210   -->wf 5212   ` cfv 5216  recscrecs 6304  freccfrec 6390
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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4118  ax-sep 4121  ax-nul 4129  ax-pow 4174  ax-pr 4209  ax-un 4433  ax-setind 4536  ax-iinf 4587
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-int 3845  df-iun 3888  df-br 4004  df-opab 4065  df-mpt 4066  df-tr 4102  df-id 4293  df-iord 4366  df-on 4368  df-ilim 4369  df-suc 4371  df-iom 4590  df-xp 4632  df-rel 4633  df-cnv 4634  df-co 4635  df-dm 4636  df-rn 4637  df-res 4638  df-ima 4639  df-iota 5178  df-fun 5218  df-fn 5219  df-f 5220  df-f1 5221  df-fo 5222  df-f1o 5223  df-fv 5224  df-recs 6305  df-frec 6391
This theorem is referenced by:  freccl  6403
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