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

Theorem freccllem 6416
Description: Lemma for freccl 6417. 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 6405 . . . 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 4915 . . . 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 2211 . . 3  |- frec ( F ,  A )  =  ( G  |`  om )
54fveq1i 5528 . 2  |-  (frec ( F ,  A ) `
 B )  =  ( ( G  |`  om ) `  B )
6 freccl.b . . . 4  |-  ( ph  ->  B  e.  om )
7 fvres 5551 . . . 4  |-  ( B  e.  om  ->  (
( G  |`  om ) `  B )  =  ( G `  B ) )
86, 7syl 14 . . 3  |-  ( ph  ->  ( ( G  |`  om ) `  B )  =  ( G `  B ) )
9 funmpt 5266 . . . . 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 4618 . . . . 5  |-  Ord  om
1211a1i 9 . . . 4  |-  ( ph  ->  Ord  om )
13 vex 2752 . . . . . 6  |-  f  e. 
_V
14 simp2 999 . . . . . . 7  |-  ( (
ph  /\  y  e.  om 
/\  f : y --> S )  ->  y  e.  om )
15 simp3 1000 . . . . . . 7  |-  ( (
ph  /\  y  e.  om 
/\  f : y --> S )  ->  f : y --> S )
16 freccl.cl . . . . . . . . 9  |-  ( (
ph  /\  z  e.  S )  ->  ( F `  z )  e.  S )
1716ralrimiva 2560 . . . . . . . 8  |-  ( ph  ->  A. z  e.  S  ( F `  z )  e.  S )
18173ad2ant1 1019 . . . . . . 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 1019 . . . . . . 7  |-  ( (
ph  /\  y  e.  om 
/\  f : y --> S )  ->  A  e.  S )
2114, 15, 18, 20frecabcl 6413 . . . . . 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 4839 . . . . . . . . . . . 12  |-  ( g  =  f  ->  dom  g  =  dom  f )
2322eqeq1d 2196 . . . . . . . . . . 11  |-  ( g  =  f  ->  ( dom  g  =  suc  m 
<->  dom  f  =  suc  m ) )
24 fveq1 5526 . . . . . . . . . . . . 13  |-  ( g  =  f  ->  (
g `  m )  =  ( f `  m ) )
2524fveq2d 5531 . . . . . . . . . . . 12  |-  ( g  =  f  ->  ( F `  ( g `  m ) )  =  ( F `  (
f `  m )
) )
2625eleq2d 2257 . . . . . . . . . . 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 2488 . . . . . . . . 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 2196 . . . . . . . . . 10  |-  ( g  =  f  ->  ( dom  g  =  (/)  <->  dom  f  =  (/) ) )
3029anbi1d 465 . . . . . . . . 9  |-  ( g  =  f  ->  (
( dom  g  =  (/) 
/\  x  e.  A
)  <->  ( dom  f  =  (/)  /\  x  e.  A ) ) )
3128, 30orbi12d 794 . . . . . . . 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 2305 . . . . . . 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 2187 . . . . . . 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 5605 . . . . . 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 2264 . . . 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 4625 . . . . . . 7  |-  Lim  om
38 limuni 4408 . . . . . . 7  |-  ( Lim 
om  ->  om  =  U. om )
3937, 38ax-mp 5 . . . . . 6  |-  om  =  U. om
4039eleq2i 2254 . . . . 5  |-  ( y  e.  om  <->  y  e.  U.
om )
41 peano2 4606 . . . . . 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 2280 . . . 4  |-  ( ph  ->  B  e.  U. om )
452, 10, 12, 36, 43, 44tfrcl 6378 . . 3  |-  ( ph  ->  ( G `  B
)  e.  S )
468, 45eqeltrd 2264 . 2  |-  ( ph  ->  ( ( G  |`  om ) `  B )  e.  S )
475, 46eqeltrid 2274 1  |-  ( ph  ->  (frec ( F ,  A ) `  B
)  e.  S )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    \/ wo 709    /\ w3a 979    = wceq 1363    e. wcel 2158   {cab 2173   A.wral 2465   E.wrex 2466   _Vcvv 2749   (/)c0 3434   U.cuni 3821    |-> cmpt 4076   Ord word 4374   Lim wlim 4376   suc csuc 4377   omcom 4601   dom cdm 4638    |` cres 4640   Fun wfun 5222   -->wf 5224   ` cfv 5228  recscrecs 6318  freccfrec 6404
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 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-13 2160  ax-14 2161  ax-ext 2169  ax-coll 4130  ax-sep 4133  ax-nul 4141  ax-pow 4186  ax-pr 4221  ax-un 4445  ax-setind 4548  ax-iinf 4599
This theorem depends on definitions:  df-bi 117  df-3an 981  df-tru 1366  df-fal 1369  df-nf 1471  df-sb 1773  df-eu 2039  df-mo 2040  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ne 2358  df-ral 2470  df-rex 2471  df-reu 2472  df-rab 2474  df-v 2751  df-sbc 2975  df-csb 3070  df-dif 3143  df-un 3145  df-in 3147  df-ss 3154  df-nul 3435  df-pw 3589  df-sn 3610  df-pr 3611  df-op 3613  df-uni 3822  df-int 3857  df-iun 3900  df-br 4016  df-opab 4077  df-mpt 4078  df-tr 4114  df-id 4305  df-iord 4378  df-on 4380  df-ilim 4381  df-suc 4383  df-iom 4602  df-xp 4644  df-rel 4645  df-cnv 4646  df-co 4647  df-dm 4648  df-rn 4649  df-res 4650  df-ima 4651  df-iota 5190  df-fun 5230  df-fn 5231  df-f 5232  df-f1 5233  df-fo 5234  df-f1o 5235  df-fv 5236  df-recs 6319  df-frec 6405
This theorem is referenced by:  freccl  6417
  Copyright terms: Public domain W3C validator