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

Theorem frecuzrdgfunlem 10375
Description: The recursive definition generator on upper integers produces a a function. (Contributed by Jim Kingdon, 24-Apr-2022.)
Hypotheses
Ref Expression
frecuzrdgrclt.c  |-  ( ph  ->  C  e.  ZZ )
frecuzrdgrclt.a  |-  ( ph  ->  A  e.  S )
frecuzrdgrclt.t  |-  ( ph  ->  S  C_  T )
frecuzrdgrclt.f  |-  ( (
ph  /\  ( x  e.  ( ZZ>= `  C )  /\  y  e.  S
) )  ->  (
x F y )  e.  S )
frecuzrdgrclt.r  |-  R  = frec ( ( x  e.  ( ZZ>= `  C ) ,  y  e.  T  |-> 
<. ( x  +  1 ) ,  ( x F y ) >.
) ,  <. C ,  A >. )
frecuzrdgfunlem.g  |-  G  = frec ( ( x  e.  ZZ  |->  ( x  + 
1 ) ) ,  C )
Assertion
Ref Expression
frecuzrdgfunlem  |-  ( ph  ->  Fun  ran  R )
Distinct variable groups:    x, C, y   
x, F, y    x, S, y    x, T, y    ph, x, y    x, G, y    x, R, y
Allowed substitution hints:    A( x, y)

Proof of Theorem frecuzrdgfunlem
Dummy variables  z  w  v are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 frecuzrdgrclt.c . . . . . 6  |-  ( ph  ->  C  e.  ZZ )
2 frecuzrdgrclt.a . . . . . 6  |-  ( ph  ->  A  e.  S )
3 frecuzrdgrclt.t . . . . . 6  |-  ( ph  ->  S  C_  T )
4 frecuzrdgrclt.f . . . . . 6  |-  ( (
ph  /\  ( x  e.  ( ZZ>= `  C )  /\  y  e.  S
) )  ->  (
x F y )  e.  S )
5 frecuzrdgrclt.r . . . . . 6  |-  R  = frec ( ( x  e.  ( ZZ>= `  C ) ,  y  e.  T  |-> 
<. ( x  +  1 ) ,  ( x F y ) >.
) ,  <. C ,  A >. )
61, 2, 3, 4, 5frecuzrdgrclt 10371 . . . . 5  |-  ( ph  ->  R : om --> ( (
ZZ>= `  C )  X.  S ) )
7 frn 5356 . . . . 5  |-  ( R : om --> ( (
ZZ>= `  C )  X.  S )  ->  ran  R 
C_  ( ( ZZ>= `  C )  X.  S
) )
86, 7syl 14 . . . 4  |-  ( ph  ->  ran  R  C_  (
( ZZ>= `  C )  X.  S ) )
9 xpss 4719 . . . 4  |-  ( (
ZZ>= `  C )  X.  S )  C_  ( _V  X.  _V )
108, 9sstrdi 3159 . . 3  |-  ( ph  ->  ran  R  C_  ( _V  X.  _V ) )
11 df-rel 4618 . . 3  |-  ( Rel 
ran  R  <->  ran  R  C_  ( _V  X.  _V ) )
1210, 11sylibr 133 . 2  |-  ( ph  ->  Rel  ran  R )
13 frecuzrdgfunlem.g . . . . . . . . . 10  |-  G  = frec ( ( x  e.  ZZ  |->  ( x  + 
1 ) ) ,  C )
141, 13frec2uzf1od 10362 . . . . . . . . 9  |-  ( ph  ->  G : om -1-1-onto-> ( ZZ>= `  C )
)
15 f1ocnvdm 5760 . . . . . . . . 9  |-  ( ( G : om -1-1-onto-> ( ZZ>= `  C )  /\  v  e.  ( ZZ>=
`  C ) )  ->  ( `' G `  v )  e.  om )
1614, 15sylan 281 . . . . . . . 8  |-  ( (
ph  /\  v  e.  ( ZZ>= `  C )
)  ->  ( `' G `  v )  e.  om )
176ffvelrnda 5631 . . . . . . . 8  |-  ( (
ph  /\  ( `' G `  v )  e.  om )  ->  ( R `  ( `' G `  v )
)  e.  ( (
ZZ>= `  C )  X.  S ) )
1816, 17syldan 280 . . . . . . 7  |-  ( (
ph  /\  v  e.  ( ZZ>= `  C )
)  ->  ( R `  ( `' G `  v ) )  e.  ( ( ZZ>= `  C
)  X.  S ) )
19 xp2nd 6145 . . . . . . 7  |-  ( ( R `  ( `' G `  v ) )  e.  ( (
ZZ>= `  C )  X.  S )  ->  ( 2nd `  ( R `  ( `' G `  v ) ) )  e.  S
)
2018, 19syl 14 . . . . . 6  |-  ( (
ph  /\  v  e.  ( ZZ>= `  C )
)  ->  ( 2nd `  ( R `  ( `' G `  v ) ) )  e.  S
)
21 ffn 5347 . . . . . . . . . 10  |-  ( R : om --> ( (
ZZ>= `  C )  X.  S )  ->  R  Fn  om )
22 fvelrnb 5544 . . . . . . . . . 10  |-  ( R  Fn  om  ->  ( <. v ,  z >.  e.  ran  R  <->  E. w  e.  om  ( R `  w )  =  <. v ,  z >. )
)
236, 21, 223syl 17 . . . . . . . . 9  |-  ( ph  ->  ( <. v ,  z
>.  e.  ran  R  <->  E. w  e.  om  ( R `  w )  =  <. v ,  z >. )
)
246ffvelrnda 5631 . . . . . . . . . . . . . . . . . . 19  |-  ( (
ph  /\  w  e.  om )  ->  ( R `  w )  e.  ( ( ZZ>= `  C )  X.  S ) )
25 1st2nd2 6154 . . . . . . . . . . . . . . . . . . 19  |-  ( ( R `  w )  e.  ( ( ZZ>= `  C )  X.  S
)  ->  ( R `  w )  =  <. ( 1st `  ( R `
 w ) ) ,  ( 2nd `  ( R `  w )
) >. )
2624, 25syl 14 . . . . . . . . . . . . . . . . . 18  |-  ( (
ph  /\  w  e.  om )  ->  ( R `  w )  =  <. ( 1st `  ( R `
 w ) ) ,  ( 2nd `  ( R `  w )
) >. )
271adantr 274 . . . . . . . . . . . . . . . . . . . 20  |-  ( (
ph  /\  w  e.  om )  ->  C  e.  ZZ )
282adantr 274 . . . . . . . . . . . . . . . . . . . 20  |-  ( (
ph  /\  w  e.  om )  ->  A  e.  S )
293adantr 274 . . . . . . . . . . . . . . . . . . . 20  |-  ( (
ph  /\  w  e.  om )  ->  S  C_  T
)
304adantlr 474 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( ph  /\  w  e.  om )  /\  (
x  e.  ( ZZ>= `  C )  /\  y  e.  S ) )  -> 
( x F y )  e.  S )
31 simpr 109 . . . . . . . . . . . . . . . . . . . 20  |-  ( (
ph  /\  w  e.  om )  ->  w  e.  om )
3227, 28, 29, 30, 5, 31, 13frecuzrdgg 10372 . . . . . . . . . . . . . . . . . . 19  |-  ( (
ph  /\  w  e.  om )  ->  ( 1st `  ( R `  w
) )  =  ( G `  w ) )
3332opeq1d 3771 . . . . . . . . . . . . . . . . . 18  |-  ( (
ph  /\  w  e.  om )  ->  <. ( 1st `  ( R `  w
) ) ,  ( 2nd `  ( R `
 w ) )
>.  =  <. ( G `
 w ) ,  ( 2nd `  ( R `  w )
) >. )
3426, 33eqtrd 2203 . . . . . . . . . . . . . . . . 17  |-  ( (
ph  /\  w  e.  om )  ->  ( R `  w )  =  <. ( G `  w ) ,  ( 2nd `  ( R `  w )
) >. )
3534eqeq1d 2179 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  w  e.  om )  ->  ( ( R `  w )  =  <. v ,  z
>. 
<-> 
<. ( G `  w
) ,  ( 2nd `  ( R `  w
) ) >.  =  <. v ,  z >. )
)
36 vex 2733 . . . . . . . . . . . . . . . . . 18  |-  v  e. 
_V
37 vex 2733 . . . . . . . . . . . . . . . . . 18  |-  z  e. 
_V
3836, 37opth2 4225 . . . . . . . . . . . . . . . . 17  |-  ( <.
( G `  w
) ,  ( 2nd `  ( R `  w
) ) >.  =  <. v ,  z >.  <->  ( ( G `  w )  =  v  /\  ( 2nd `  ( R `  w ) )  =  z ) )
3938simplbi 272 . . . . . . . . . . . . . . . 16  |-  ( <.
( G `  w
) ,  ( 2nd `  ( R `  w
) ) >.  =  <. v ,  z >.  ->  ( G `  w )  =  v )
4035, 39syl6bi 162 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  w  e.  om )  ->  ( ( R `  w )  =  <. v ,  z
>.  ->  ( G `  w )  =  v ) )
41 f1ocnvfv 5758 . . . . . . . . . . . . . . . 16  |-  ( ( G : om -1-1-onto-> ( ZZ>= `  C )  /\  w  e.  om )  ->  ( ( G `
 w )  =  v  ->  ( `' G `  v )  =  w ) )
4214, 41sylan 281 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  w  e.  om )  ->  ( ( G `  w )  =  v  ->  ( `' G `  v )  =  w ) )
4340, 42syld 45 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  w  e.  om )  ->  ( ( R `  w )  =  <. v ,  z
>.  ->  ( `' G `  v )  =  w ) )
44 fveq2 5496 . . . . . . . . . . . . . . 15  |-  ( ( `' G `  v )  =  w  ->  ( R `  ( `' G `  v )
)  =  ( R `
 w ) )
4544fveq2d 5500 . . . . . . . . . . . . . 14  |-  ( ( `' G `  v )  =  w  ->  ( 2nd `  ( R `  ( `' G `  v ) ) )  =  ( 2nd `  ( R `
 w ) ) )
4643, 45syl6 33 . . . . . . . . . . . . 13  |-  ( (
ph  /\  w  e.  om )  ->  ( ( R `  w )  =  <. v ,  z
>.  ->  ( 2nd `  ( R `  ( `' G `  v )
) )  =  ( 2nd `  ( R `
 w ) ) ) )
4746imp 123 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  w  e.  om )  /\  ( R `  w )  =  <. v ,  z
>. )  ->  ( 2nd `  ( R `  ( `' G `  v ) ) )  =  ( 2nd `  ( R `
 w ) ) )
4836, 37op2ndd 6128 . . . . . . . . . . . . 13  |-  ( ( R `  w )  =  <. v ,  z
>.  ->  ( 2nd `  ( R `  w )
)  =  z )
4948adantl 275 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  w  e.  om )  /\  ( R `  w )  =  <. v ,  z
>. )  ->  ( 2nd `  ( R `  w
) )  =  z )
5047, 49eqtr2d 2204 . . . . . . . . . . 11  |-  ( ( ( ph  /\  w  e.  om )  /\  ( R `  w )  =  <. v ,  z
>. )  ->  z  =  ( 2nd `  ( R `  ( `' G `  v )
) ) )
5150ex 114 . . . . . . . . . 10  |-  ( (
ph  /\  w  e.  om )  ->  ( ( R `  w )  =  <. v ,  z
>.  ->  z  =  ( 2nd `  ( R `
 ( `' G `  v ) ) ) ) )
5251rexlimdva 2587 . . . . . . . . 9  |-  ( ph  ->  ( E. w  e. 
om  ( R `  w )  =  <. v ,  z >.  ->  z  =  ( 2nd `  ( R `  ( `' G `  v )
) ) ) )
5323, 52sylbid 149 . . . . . . . 8  |-  ( ph  ->  ( <. v ,  z
>.  e.  ran  R  -> 
z  =  ( 2nd `  ( R `  ( `' G `  v ) ) ) ) )
5453alrimiv 1867 . . . . . . 7  |-  ( ph  ->  A. z ( <.
v ,  z >.  e.  ran  R  ->  z  =  ( 2nd `  ( R `  ( `' G `  v )
) ) ) )
5554adantr 274 . . . . . 6  |-  ( (
ph  /\  v  e.  ( ZZ>= `  C )
)  ->  A. z
( <. v ,  z
>.  e.  ran  R  -> 
z  =  ( 2nd `  ( R `  ( `' G `  v ) ) ) ) )
56 eqeq2 2180 . . . . . . . . 9  |-  ( w  =  ( 2nd `  ( R `  ( `' G `  v )
) )  ->  (
z  =  w  <->  z  =  ( 2nd `  ( R `
 ( `' G `  v ) ) ) ) )
5756imbi2d 229 . . . . . . . 8  |-  ( w  =  ( 2nd `  ( R `  ( `' G `  v )
) )  ->  (
( <. v ,  z
>.  e.  ran  R  -> 
z  =  w )  <-> 
( <. v ,  z
>.  e.  ran  R  -> 
z  =  ( 2nd `  ( R `  ( `' G `  v ) ) ) ) ) )
5857albidv 1817 . . . . . . 7  |-  ( w  =  ( 2nd `  ( R `  ( `' G `  v )
) )  ->  ( A. z ( <. v ,  z >.  e.  ran  R  ->  z  =  w )  <->  A. z ( <.
v ,  z >.  e.  ran  R  ->  z  =  ( 2nd `  ( R `  ( `' G `  v )
) ) ) ) )
5958spcegv 2818 . . . . . 6  |-  ( ( 2nd `  ( R `
 ( `' G `  v ) ) )  e.  S  ->  ( A. z ( <. v ,  z >.  e.  ran  R  ->  z  =  ( 2nd `  ( R `
 ( `' G `  v ) ) ) )  ->  E. w A. z ( <. v ,  z >.  e.  ran  R  ->  z  =  w ) ) )
6020, 55, 59sylc 62 . . . . 5  |-  ( (
ph  /\  v  e.  ( ZZ>= `  C )
)  ->  E. w A. z ( <. v ,  z >.  e.  ran  R  ->  z  =  w ) )
61 nfv 1521 . . . . . 6  |-  F/ w <. v ,  z >.  e.  ran  R
6261mo2r 2071 . . . . 5  |-  ( E. w A. z (
<. v ,  z >.  e.  ran  R  ->  z  =  w )  ->  E* z <. v ,  z
>.  e.  ran  R )
6360, 62syl 14 . . . 4  |-  ( (
ph  /\  v  e.  ( ZZ>= `  C )
)  ->  E* z <. v ,  z >.  e.  ran  R )
641, 2, 3, 4, 5frecuzrdgdom 10374 . . . . . 6  |-  ( ph  ->  dom  ran  R  =  ( ZZ>= `  C )
)
6564eleq2d 2240 . . . . 5  |-  ( ph  ->  ( v  e.  dom  ran 
R  <->  v  e.  (
ZZ>= `  C ) ) )
6665pm5.32i 451 . . . 4  |-  ( (
ph  /\  v  e.  dom  ran  R )  <->  ( ph  /\  v  e.  ( ZZ>= `  C ) ) )
67 df-br 3990 . . . . 5  |-  ( v ran  R  z  <->  <. v ,  z >.  e.  ran  R )
6867mobii 2056 . . . 4  |-  ( E* z  v ran  R  z 
<->  E* z <. v ,  z >.  e.  ran  R )
6963, 66, 683imtr4i 200 . . 3  |-  ( (
ph  /\  v  e.  dom  ran  R )  ->  E* z  v ran  R  z )
7069ralrimiva 2543 . 2  |-  ( ph  ->  A. v  e.  dom  ran 
R E* z  v ran  R  z )
71 dffun7 5225 . 2  |-  ( Fun 
ran  R  <->  ( Rel  ran  R  /\  A. v  e. 
dom  ran  R E* z 
v ran  R  z
) )
7212, 70, 71sylanbrc 415 1  |-  ( ph  ->  Fun  ran  R )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104   A.wal 1346    = wceq 1348   E.wex 1485   E*wmo 2020    e. wcel 2141   A.wral 2448   E.wrex 2449   _Vcvv 2730    C_ wss 3121   <.cop 3586   class class class wbr 3989    |-> cmpt 4050   omcom 4574    X. cxp 4609   `'ccnv 4610   dom cdm 4611   ran crn 4612   Rel wrel 4616   Fun wfun 5192    Fn wfn 5193   -->wf 5194   -1-1-onto->wf1o 5197   ` cfv 5198  (class class class)co 5853    e. cmpo 5855   1stc1st 6117   2ndc2nd 6118  freccfrec 6369   1c1 7775    + caddc 7777   ZZcz 9212   ZZ>=cuz 9487
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-coll 4104  ax-sep 4107  ax-nul 4115  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521  ax-iinf 4572  ax-cnex 7865  ax-resscn 7866  ax-1cn 7867  ax-1re 7868  ax-icn 7869  ax-addcl 7870  ax-addrcl 7871  ax-mulcl 7872  ax-addcom 7874  ax-addass 7876  ax-distr 7878  ax-i2m1 7879  ax-0lt1 7880  ax-0id 7882  ax-rnegex 7883  ax-cnre 7885  ax-pre-ltirr 7886  ax-pre-ltwlin 7887  ax-pre-lttrn 7888  ax-pre-ltadd 7890
This theorem depends on definitions:  df-bi 116  df-3or 974  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-nel 2436  df-ral 2453  df-rex 2454  df-reu 2455  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-int 3832  df-iun 3875  df-br 3990  df-opab 4051  df-mpt 4052  df-tr 4088  df-id 4278  df-iord 4351  df-on 4353  df-ilim 4354  df-suc 4356  df-iom 4575  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-f1 5203  df-fo 5204  df-f1o 5205  df-fv 5206  df-riota 5809  df-ov 5856  df-oprab 5857  df-mpo 5858  df-1st 6119  df-2nd 6120  df-recs 6284  df-frec 6370  df-pnf 7956  df-mnf 7957  df-xr 7958  df-ltxr 7959  df-le 7960  df-sub 8092  df-neg 8093  df-inn 8879  df-n0 9136  df-z 9213  df-uz 9488
This theorem is referenced by:  frecuzrdgfun  10376
  Copyright terms: Public domain W3C validator