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

Theorem rdgon 6003
Description: Evaluating the recursive definition generator produces an ordinal. There is a hypothesis that the characteristic function produces ordinals on ordinal arguments. (Contributed by Jim Kingdon, 26-Jul-2019.)
Hypotheses
Ref Expression
rdgon.1  |-  ( ph  ->  F  Fn  _V )
rdgon.2  |-  ( ph  ->  A  e.  On )
rdgon.3  |-  ( ph  ->  A. x  e.  On  ( F `  x )  e.  On )
Assertion
Ref Expression
rdgon  |-  ( (
ph  /\  B  e.  On )  ->  ( rec ( F ,  A
) `  B )  e.  On )
Distinct variable groups:    x, A    x, F    ph, x
Allowed substitution hint:    B( x)

Proof of Theorem rdgon
Dummy variables  w  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fveq2 5205 . . . . 5  |-  ( z  =  x  ->  ( rec ( F ,  A
) `  z )  =  ( rec ( F ,  A ) `  x ) )
21eleq1d 2122 . . . 4  |-  ( z  =  x  ->  (
( rec ( F ,  A ) `  z )  e.  On  <->  ( rec ( F ,  A ) `  x
)  e.  On ) )
32imbi2d 223 . . 3  |-  ( z  =  x  ->  (
( ph  ->  ( rec ( F ,  A
) `  z )  e.  On )  <->  ( ph  ->  ( rec ( F ,  A ) `  x )  e.  On ) ) )
4 fveq2 5205 . . . . 5  |-  ( z  =  B  ->  ( rec ( F ,  A
) `  z )  =  ( rec ( F ,  A ) `  B ) )
54eleq1d 2122 . . . 4  |-  ( z  =  B  ->  (
( rec ( F ,  A ) `  z )  e.  On  <->  ( rec ( F ,  A ) `  B
)  e.  On ) )
65imbi2d 223 . . 3  |-  ( z  =  B  ->  (
( ph  ->  ( rec ( F ,  A
) `  z )  e.  On )  <->  ( ph  ->  ( rec ( F ,  A ) `  B )  e.  On ) ) )
7 r19.21v 2413 . . . 4  |-  ( A. x  e.  z  ( ph  ->  ( rec ( F ,  A ) `  x )  e.  On ) 
<->  ( ph  ->  A. x  e.  z  ( rec ( F ,  A ) `
 x )  e.  On ) )
8 rdgon.2 . . . . . . . . 9  |-  ( ph  ->  A  e.  On )
9 fvres 5225 . . . . . . . . . . . . . 14  |-  ( x  e.  z  ->  (
( rec ( F ,  A )  |`  z ) `  x
)  =  ( rec ( F ,  A
) `  x )
)
109eleq1d 2122 . . . . . . . . . . . . 13  |-  ( x  e.  z  ->  (
( ( rec ( F ,  A )  |`  z ) `  x
)  e.  On  <->  ( rec ( F ,  A ) `
 x )  e.  On ) )
1110adantl 266 . . . . . . . . . . . 12  |-  ( (
ph  /\  x  e.  z )  ->  (
( ( rec ( F ,  A )  |`  z ) `  x
)  e.  On  <->  ( rec ( F ,  A ) `
 x )  e.  On ) )
12 rdgon.3 . . . . . . . . . . . . . . 15  |-  ( ph  ->  A. x  e.  On  ( F `  x )  e.  On )
13 fveq2 5205 . . . . . . . . . . . . . . . . 17  |-  ( x  =  w  ->  ( F `  x )  =  ( F `  w ) )
1413eleq1d 2122 . . . . . . . . . . . . . . . 16  |-  ( x  =  w  ->  (
( F `  x
)  e.  On  <->  ( F `  w )  e.  On ) )
1514cbvralv 2550 . . . . . . . . . . . . . . 15  |-  ( A. x  e.  On  ( F `  x )  e.  On  <->  A. w  e.  On  ( F `  w )  e.  On )
1612, 15sylib 131 . . . . . . . . . . . . . 14  |-  ( ph  ->  A. w  e.  On  ( F `  w )  e.  On )
17 fveq2 5205 . . . . . . . . . . . . . . . 16  |-  ( w  =  ( ( rec ( F ,  A
)  |`  z ) `  x )  ->  ( F `  w )  =  ( F `  ( ( rec ( F ,  A )  |`  z ) `  x
) ) )
1817eleq1d 2122 . . . . . . . . . . . . . . 15  |-  ( w  =  ( ( rec ( F ,  A
)  |`  z ) `  x )  ->  (
( F `  w
)  e.  On  <->  ( F `  ( ( rec ( F ,  A )  |`  z ) `  x
) )  e.  On ) )
1918rspcv 2669 . . . . . . . . . . . . . 14  |-  ( ( ( rec ( F ,  A )  |`  z ) `  x
)  e.  On  ->  ( A. w  e.  On  ( F `  w )  e.  On  ->  ( F `  ( ( rec ( F ,  A
)  |`  z ) `  x ) )  e.  On ) )
2016, 19syl5com 29 . . . . . . . . . . . . 13  |-  ( ph  ->  ( ( ( rec ( F ,  A
)  |`  z ) `  x )  e.  On  ->  ( F `  (
( rec ( F ,  A )  |`  z ) `  x
) )  e.  On ) )
2120adantr 265 . . . . . . . . . . . 12  |-  ( (
ph  /\  x  e.  z )  ->  (
( ( rec ( F ,  A )  |`  z ) `  x
)  e.  On  ->  ( F `  ( ( rec ( F ,  A )  |`  z
) `  x )
)  e.  On ) )
2211, 21sylbird 163 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  z )  ->  (
( rec ( F ,  A ) `  x )  e.  On  ->  ( F `  (
( rec ( F ,  A )  |`  z ) `  x
) )  e.  On ) )
2322ralimdva 2404 . . . . . . . . . 10  |-  ( ph  ->  ( A. x  e.  z  ( rec ( F ,  A ) `  x )  e.  On  ->  A. x  e.  z  ( F `  (
( rec ( F ,  A )  |`  z ) `  x
) )  e.  On ) )
24 vex 2577 . . . . . . . . . . 11  |-  z  e. 
_V
25 iunon 5929 . . . . . . . . . . 11  |-  ( ( z  e.  _V  /\  A. x  e.  z  ( F `  ( ( rec ( F ,  A )  |`  z
) `  x )
)  e.  On )  ->  U_ x  e.  z  ( F `  (
( rec ( F ,  A )  |`  z ) `  x
) )  e.  On )
2624, 25mpan 408 . . . . . . . . . 10  |-  ( A. x  e.  z  ( F `  ( ( rec ( F ,  A
)  |`  z ) `  x ) )  e.  On  ->  U_ x  e.  z  ( F `  ( ( rec ( F ,  A )  |`  z ) `  x
) )  e.  On )
2723, 26syl6 33 . . . . . . . . 9  |-  ( ph  ->  ( A. x  e.  z  ( rec ( F ,  A ) `  x )  e.  On  ->  U_ x  e.  z  ( F `  (
( rec ( F ,  A )  |`  z ) `  x
) )  e.  On ) )
28 onun2 4243 . . . . . . . . 9  |-  ( ( A  e.  On  /\  U_ x  e.  z  ( F `  ( ( rec ( F ,  A )  |`  z
) `  x )
)  e.  On )  ->  ( A  u.  U_ x  e.  z  ( F `  ( ( rec ( F ,  A )  |`  z
) `  x )
) )  e.  On )
298, 27, 28syl6an 1339 . . . . . . . 8  |-  ( ph  ->  ( A. x  e.  z  ( rec ( F ,  A ) `  x )  e.  On  ->  ( A  u.  U_ x  e.  z  ( F `  ( ( rec ( F ,  A
)  |`  z ) `  x ) ) )  e.  On ) )
3029adantr 265 . . . . . . 7  |-  ( (
ph  /\  z  e.  On )  ->  ( A. x  e.  z  ( rec ( F ,  A
) `  x )  e.  On  ->  ( A  u.  U_ x  e.  z  ( F `  (
( rec ( F ,  A )  |`  z ) `  x
) ) )  e.  On ) )
31 rdgon.1 . . . . . . . . . 10  |-  ( ph  ->  F  Fn  _V )
3231, 8jca 294 . . . . . . . . 9  |-  ( ph  ->  ( F  Fn  _V  /\  A  e.  On ) )
33 rdgivallem 5998 . . . . . . . . . 10  |-  ( ( F  Fn  _V  /\  A  e.  On  /\  z  e.  On )  ->  ( rec ( F ,  A
) `  z )  =  ( A  u.  U_ x  e.  z  ( F `  ( ( rec ( F ,  A )  |`  z
) `  x )
) ) )
34333expa 1115 . . . . . . . . 9  |-  ( ( ( F  Fn  _V  /\  A  e.  On )  /\  z  e.  On )  ->  ( rec ( F ,  A ) `  z )  =  ( A  u.  U_ x  e.  z  ( F `  ( ( rec ( F ,  A )  |`  z ) `  x
) ) ) )
3532, 34sylan 271 . . . . . . . 8  |-  ( (
ph  /\  z  e.  On )  ->  ( rec ( F ,  A
) `  z )  =  ( A  u.  U_ x  e.  z  ( F `  ( ( rec ( F ,  A )  |`  z
) `  x )
) ) )
3635eleq1d 2122 . . . . . . 7  |-  ( (
ph  /\  z  e.  On )  ->  ( ( rec ( F ,  A ) `  z
)  e.  On  <->  ( A  u.  U_ x  e.  z  ( F `  (
( rec ( F ,  A )  |`  z ) `  x
) ) )  e.  On ) )
3730, 36sylibrd 162 . . . . . 6  |-  ( (
ph  /\  z  e.  On )  ->  ( A. x  e.  z  ( rec ( F ,  A
) `  x )  e.  On  ->  ( rec ( F ,  A ) `
 z )  e.  On ) )
3837expcom 113 . . . . 5  |-  ( z  e.  On  ->  ( ph  ->  ( A. x  e.  z  ( rec ( F ,  A ) `
 x )  e.  On  ->  ( rec ( F ,  A ) `
 z )  e.  On ) ) )
3938a2d 26 . . . 4  |-  ( z  e.  On  ->  (
( ph  ->  A. x  e.  z  ( rec ( F ,  A ) `
 x )  e.  On )  ->  ( ph  ->  ( rec ( F ,  A ) `  z )  e.  On ) ) )
407, 39syl5bi 145 . . 3  |-  ( z  e.  On  ->  ( A. x  e.  z 
( ph  ->  ( rec ( F ,  A
) `  x )  e.  On )  ->  ( ph  ->  ( rec ( F ,  A ) `  z )  e.  On ) ) )
413, 6, 40tfis3 4336 . 2  |-  ( B  e.  On  ->  ( ph  ->  ( rec ( F ,  A ) `  B )  e.  On ) )
4241impcom 120 1  |-  ( (
ph  /\  B  e.  On )  ->  ( rec ( F ,  A
) `  B )  e.  On )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 101    <-> wb 102    = wceq 1259    e. wcel 1409   A.wral 2323   _Vcvv 2574    u. cun 2942   U_ciun 3684   Oncon0 4127    |` cres 4374    Fn wfn 4924   ` cfv 4929   reccrdg 5986
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-13 1420  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-coll 3899  ax-sep 3902  ax-pow 3954  ax-pr 3971  ax-un 4197  ax-setind 4289
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-fal 1265  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ne 2221  df-ral 2328  df-rex 2329  df-reu 2330  df-rab 2332  df-v 2576  df-sbc 2787  df-csb 2880  df-dif 2947  df-un 2949  df-in 2951  df-ss 2958  df-nul 3252  df-pw 3388  df-sn 3408  df-pr 3409  df-op 3411  df-uni 3608  df-iun 3686  df-br 3792  df-opab 3846  df-mpt 3847  df-tr 3882  df-id 4057  df-iord 4130  df-on 4132  df-suc 4135  df-xp 4378  df-rel 4379  df-cnv 4380  df-co 4381  df-dm 4382  df-rn 4383  df-res 4384  df-ima 4385  df-iota 4894  df-fun 4931  df-fn 4932  df-f 4933  df-f1 4934  df-fo 4935  df-f1o 4936  df-fv 4937  df-recs 5950  df-irdg 5987
This theorem is referenced by:  oacl  6070  omcl  6071  oeicl  6072
  Copyright terms: Public domain W3C validator