ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  freccllem Unicode version

Theorem freccllem 6251
Description: Lemma for freccl 6252. 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 6240 . . . 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 4771 . . . 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 2136 . . 3  |- frec ( F ,  A )  =  ( G  |`  om )
54fveq1i 5374 . 2  |-  (frec ( F ,  A ) `
 B )  =  ( ( G  |`  om ) `  B )
6 freccl.b . . . 4  |-  ( ph  ->  B  e.  om )
7 fvres 5397 . . . 4  |-  ( B  e.  om  ->  (
( G  |`  om ) `  B )  =  ( G `  B ) )
86, 7syl 14 . . 3  |-  ( ph  ->  ( ( G  |`  om ) `  B )  =  ( G `  B ) )
9 funmpt 5117 . . . . 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 4478 . . . . 5  |-  Ord  om
1211a1i 9 . . . 4  |-  ( ph  ->  Ord  om )
13 vex 2658 . . . . . 6  |-  f  e. 
_V
14 simp2 963 . . . . . . 7  |-  ( (
ph  /\  y  e.  om 
/\  f : y --> S )  ->  y  e.  om )
15 simp3 964 . . . . . . 7  |-  ( (
ph  /\  y  e.  om 
/\  f : y --> S )  ->  f : y --> S )
16 freccl.cl . . . . . . . . 9  |-  ( (
ph  /\  z  e.  S )  ->  ( F `  z )  e.  S )
1716ralrimiva 2477 . . . . . . . 8  |-  ( ph  ->  A. z  e.  S  ( F `  z )  e.  S )
18173ad2ant1 983 . . . . . . 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 983 . . . . . . 7  |-  ( (
ph  /\  y  e.  om 
/\  f : y --> S )  ->  A  e.  S )
2114, 15, 18, 20frecabcl 6248 . . . . . 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 4697 . . . . . . . . . . . 12  |-  ( g  =  f  ->  dom  g  =  dom  f )
2322eqeq1d 2121 . . . . . . . . . . 11  |-  ( g  =  f  ->  ( dom  g  =  suc  m 
<->  dom  f  =  suc  m ) )
24 fveq1 5372 . . . . . . . . . . . . 13  |-  ( g  =  f  ->  (
g `  m )  =  ( f `  m ) )
2524fveq2d 5377 . . . . . . . . . . . 12  |-  ( g  =  f  ->  ( F `  ( g `  m ) )  =  ( F `  (
f `  m )
) )
2625eleq2d 2182 . . . . . . . . . . 11  |-  ( g  =  f  ->  (
x  e.  ( F `
 ( g `  m ) )  <->  x  e.  ( F `  ( f `
 m ) ) ) )
2723, 26anbi12d 462 . . . . . . . . . 10  |-  ( g  =  f  ->  (
( dom  g  =  suc  m  /\  x  e.  ( F `  (
g `  m )
) )  <->  ( dom  f  =  suc  m  /\  x  e.  ( F `  ( f `  m
) ) ) ) )
2827rexbidv 2410 . . . . . . . . 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 2121 . . . . . . . . . 10  |-  ( g  =  f  ->  ( dom  g  =  (/)  <->  dom  f  =  (/) ) )
3029anbi1d 458 . . . . . . . . 9  |-  ( g  =  f  ->  (
( dom  g  =  (/) 
/\  x  e.  A
)  <->  ( dom  f  =  (/)  /\  x  e.  A ) ) )
3128, 30orbi12d 765 . . . . . . . 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 2230 . . . . . . 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 2113 . . . . . . 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 5449 . . . . . 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 408 . . . . 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 2189 . . . 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 4485 . . . . . . 7  |-  Lim  om
38 limuni 4276 . . . . . . 7  |-  ( Lim 
om  ->  om  =  U. om )
3937, 38ax-mp 7 . . . . . 6  |-  om  =  U. om
4039eleq2i 2179 . . . . 5  |-  ( y  e.  om  <->  y  e.  U.
om )
41 peano2 4467 . . . . . 6  |-  ( y  e.  om  ->  suc  y  e.  om )
4241adantl 273 . . . . 5  |-  ( (
ph  /\  y  e.  om )  ->  suc  y  e. 
om )
4340, 42sylan2br 284 . . . 4  |-  ( (
ph  /\  y  e.  U.
om )  ->  suc  y  e.  om )
446, 39syl6eleq 2205 . . . 4  |-  ( ph  ->  B  e.  U. om )
452, 10, 12, 36, 43, 44tfrcl 6213 . . 3  |-  ( ph  ->  ( G `  B
)  e.  S )
468, 45eqeltrd 2189 . 2  |-  ( ph  ->  ( ( G  |`  om ) `  B )  e.  S )
475, 46syl5eqel 2199 1  |-  ( ph  ->  (frec ( F ,  A ) `  B
)  e.  S )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    \/ wo 680    /\ w3a 943    = wceq 1312    e. wcel 1461   {cab 2099   A.wral 2388   E.wrex 2389   _Vcvv 2655   (/)c0 3327   U.cuni 3700    |-> cmpt 3947   Ord word 4242   Lim wlim 4244   suc csuc 4245   omcom 4462   dom cdm 4497    |` cres 4499   Fun wfun 5073   -->wf 5075   ` cfv 5079  recscrecs 6153  freccfrec 6239
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 586  ax-in2 587  ax-io 681  ax-5 1404  ax-7 1405  ax-gen 1406  ax-ie1 1450  ax-ie2 1451  ax-8 1463  ax-10 1464  ax-11 1465  ax-i12 1466  ax-bndl 1467  ax-4 1468  ax-13 1472  ax-14 1473  ax-17 1487  ax-i9 1491  ax-ial 1495  ax-i5r 1496  ax-ext 2095  ax-coll 4001  ax-sep 4004  ax-nul 4012  ax-pow 4056  ax-pr 4089  ax-un 4313  ax-setind 4410  ax-iinf 4460
This theorem depends on definitions:  df-bi 116  df-3an 945  df-tru 1315  df-fal 1318  df-nf 1418  df-sb 1717  df-eu 1976  df-mo 1977  df-clab 2100  df-cleq 2106  df-clel 2109  df-nfc 2242  df-ne 2281  df-ral 2393  df-rex 2394  df-reu 2395  df-rab 2397  df-v 2657  df-sbc 2877  df-csb 2970  df-dif 3037  df-un 3039  df-in 3041  df-ss 3048  df-nul 3328  df-pw 3476  df-sn 3497  df-pr 3498  df-op 3500  df-uni 3701  df-int 3736  df-iun 3779  df-br 3894  df-opab 3948  df-mpt 3949  df-tr 3985  df-id 4173  df-iord 4246  df-on 4248  df-ilim 4249  df-suc 4251  df-iom 4463  df-xp 4503  df-rel 4504  df-cnv 4505  df-co 4506  df-dm 4507  df-rn 4508  df-res 4509  df-ima 4510  df-iota 5044  df-fun 5081  df-fn 5082  df-f 5083  df-f1 5084  df-fo 5085  df-f1o 5086  df-fv 5087  df-recs 6154  df-frec 6240
This theorem is referenced by:  freccl  6252
  Copyright terms: Public domain W3C validator