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

Theorem frecuzrdgdomlem 9713
Description: The domain of the result of the recursive definition generator on upper integers. (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 >. )
frecuzrdgdomlem.g  |-  G  = frec ( ( x  e.  ZZ  |->  ( x  + 
1 ) ) ,  C )
Assertion
Ref Expression
frecuzrdgdomlem  |-  ( ph  ->  dom  ran  R  =  ( ZZ>= `  C )
)
Distinct variable groups:    x, C, y   
x, F, y    x, S, y    x, T, y    ph, x, y    x, R, y    x, G, y
Allowed substitution hints:    A( x, y)

Proof of Theorem frecuzrdgdomlem
Dummy variable  v is 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 9711 . . . . 5  |-  ( ph  ->  R : om --> ( (
ZZ>= `  C )  X.  S ) )
7 frn 5121 . . . . 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 dmss 4593 . . . 4  |-  ( ran 
R  C_  ( ( ZZ>=
`  C )  X.  S )  ->  dom  ran 
R  C_  dom  ( (
ZZ>= `  C )  X.  S ) )
108, 9syl 14 . . 3  |-  ( ph  ->  dom  ran  R  C_  dom  ( ( ZZ>= `  C
)  X.  S ) )
11 dmxpss 4815 . . 3  |-  dom  (
( ZZ>= `  C )  X.  S )  C_  ( ZZ>=
`  C )
1210, 11syl6ss 3022 . 2  |-  ( ph  ->  dom  ran  R  C_  ( ZZ>=
`  C ) )
138adantr 270 . . . . . . . . 9  |-  ( (
ph  /\  v  e.  ( ZZ>= `  C )
)  ->  ran  R  C_  ( ( ZZ>= `  C
)  X.  S ) )
14 ffun 5117 . . . . . . . . . . . 12  |-  ( R : om --> ( (
ZZ>= `  C )  X.  S )  ->  Fun  R )
156, 14syl 14 . . . . . . . . . . 11  |-  ( ph  ->  Fun  R )
1615adantr 270 . . . . . . . . . 10  |-  ( (
ph  /\  v  e.  ( ZZ>= `  C )
)  ->  Fun  R )
17 frecuzrdgdomlem.g . . . . . . . . . . . . 13  |-  G  = frec ( ( x  e.  ZZ  |->  ( x  + 
1 ) ) ,  C )
181, 17frec2uzf1od 9702 . . . . . . . . . . . 12  |-  ( ph  ->  G : om -1-1-onto-> ( ZZ>= `  C )
)
19 f1ocnvdm 5500 . . . . . . . . . . . 12  |-  ( ( G : om -1-1-onto-> ( ZZ>= `  C )  /\  v  e.  ( ZZ>=
`  C ) )  ->  ( `' G `  v )  e.  om )
2018, 19sylan 277 . . . . . . . . . . 11  |-  ( (
ph  /\  v  e.  ( ZZ>= `  C )
)  ->  ( `' G `  v )  e.  om )
21 fdm 5119 . . . . . . . . . . . . 13  |-  ( R : om --> ( (
ZZ>= `  C )  X.  S )  ->  dom  R  =  om )
226, 21syl 14 . . . . . . . . . . . 12  |-  ( ph  ->  dom  R  =  om )
2322adantr 270 . . . . . . . . . . 11  |-  ( (
ph  /\  v  e.  ( ZZ>= `  C )
)  ->  dom  R  =  om )
2420, 23eleqtrrd 2162 . . . . . . . . . 10  |-  ( (
ph  /\  v  e.  ( ZZ>= `  C )
)  ->  ( `' G `  v )  e.  dom  R )
25 fvelrn 5375 . . . . . . . . . 10  |-  ( ( Fun  R  /\  ( `' G `  v )  e.  dom  R )  ->  ( R `  ( `' G `  v ) )  e.  ran  R
)
2616, 24, 25syl2anc 403 . . . . . . . . 9  |-  ( (
ph  /\  v  e.  ( ZZ>= `  C )
)  ->  ( R `  ( `' G `  v ) )  e. 
ran  R )
2713, 26sseldd 3011 . . . . . . . 8  |-  ( (
ph  /\  v  e.  ( ZZ>= `  C )
)  ->  ( R `  ( `' G `  v ) )  e.  ( ( ZZ>= `  C
)  X.  S ) )
28 1st2nd2 5880 . . . . . . . 8  |-  ( ( R `  ( `' G `  v ) )  e.  ( (
ZZ>= `  C )  X.  S )  ->  ( R `  ( `' G `  v )
)  =  <. ( 1st `  ( R `  ( `' G `  v ) ) ) ,  ( 2nd `  ( R `
 ( `' G `  v ) ) )
>. )
2927, 28syl 14 . . . . . . 7  |-  ( (
ph  /\  v  e.  ( ZZ>= `  C )
)  ->  ( R `  ( `' G `  v ) )  = 
<. ( 1st `  ( R `  ( `' G `  v )
) ) ,  ( 2nd `  ( R `
 ( `' G `  v ) ) )
>. )
301adantr 270 . . . . . . . . . 10  |-  ( (
ph  /\  v  e.  ( ZZ>= `  C )
)  ->  C  e.  ZZ )
312adantr 270 . . . . . . . . . 10  |-  ( (
ph  /\  v  e.  ( ZZ>= `  C )
)  ->  A  e.  S )
323adantr 270 . . . . . . . . . 10  |-  ( (
ph  /\  v  e.  ( ZZ>= `  C )
)  ->  S  C_  T
)
334adantlr 461 . . . . . . . . . 10  |-  ( ( ( ph  /\  v  e.  ( ZZ>= `  C )
)  /\  ( x  e.  ( ZZ>= `  C )  /\  y  e.  S
) )  ->  (
x F y )  e.  S )
3430, 31, 32, 33, 5, 20, 17frecuzrdgg 9712 . . . . . . . . 9  |-  ( (
ph  /\  v  e.  ( ZZ>= `  C )
)  ->  ( 1st `  ( R `  ( `' G `  v ) ) )  =  ( G `  ( `' G `  v ) ) )
35 f1ocnvfv2 5497 . . . . . . . . . 10  |-  ( ( G : om -1-1-onto-> ( ZZ>= `  C )  /\  v  e.  ( ZZ>=
`  C ) )  ->  ( G `  ( `' G `  v ) )  =  v )
3618, 35sylan 277 . . . . . . . . 9  |-  ( (
ph  /\  v  e.  ( ZZ>= `  C )
)  ->  ( G `  ( `' G `  v ) )  =  v )
3734, 36eqtrd 2115 . . . . . . . 8  |-  ( (
ph  /\  v  e.  ( ZZ>= `  C )
)  ->  ( 1st `  ( R `  ( `' G `  v ) ) )  =  v )
3837opeq1d 3602 . . . . . . 7  |-  ( (
ph  /\  v  e.  ( ZZ>= `  C )
)  ->  <. ( 1st `  ( R `  ( `' G `  v ) ) ) ,  ( 2nd `  ( R `
 ( `' G `  v ) ) )
>.  =  <. v ,  ( 2nd `  ( R `  ( `' G `  v )
) ) >. )
3929, 38eqtrd 2115 . . . . . 6  |-  ( (
ph  /\  v  e.  ( ZZ>= `  C )
)  ->  ( R `  ( `' G `  v ) )  = 
<. v ,  ( 2nd `  ( R `  ( `' G `  v ) ) ) >. )
4039, 26eqeltrrd 2160 . . . . 5  |-  ( (
ph  /\  v  e.  ( ZZ>= `  C )
)  ->  <. v ,  ( 2nd `  ( R `  ( `' G `  v )
) ) >.  e.  ran  R )
41 simpr 108 . . . . . 6  |-  ( (
ph  /\  v  e.  ( ZZ>= `  C )
)  ->  v  e.  ( ZZ>= `  C )
)
42 xp2nd 5872 . . . . . . 7  |-  ( ( R `  ( `' G `  v ) )  e.  ( (
ZZ>= `  C )  X.  S )  ->  ( 2nd `  ( R `  ( `' G `  v ) ) )  e.  S
)
4327, 42syl 14 . . . . . 6  |-  ( (
ph  /\  v  e.  ( ZZ>= `  C )
)  ->  ( 2nd `  ( R `  ( `' G `  v ) ) )  e.  S
)
44 opeldmg 4599 . . . . . 6  |-  ( ( v  e.  ( ZZ>= `  C )  /\  ( 2nd `  ( R `  ( `' G `  v ) ) )  e.  S
)  ->  ( <. v ,  ( 2nd `  ( R `  ( `' G `  v )
) ) >.  e.  ran  R  ->  v  e.  dom  ran 
R ) )
4541, 43, 44syl2anc 403 . . . . 5  |-  ( (
ph  /\  v  e.  ( ZZ>= `  C )
)  ->  ( <. v ,  ( 2nd `  ( R `  ( `' G `  v )
) ) >.  e.  ran  R  ->  v  e.  dom  ran 
R ) )
4640, 45mpd 13 . . . 4  |-  ( (
ph  /\  v  e.  ( ZZ>= `  C )
)  ->  v  e.  dom  ran  R )
4746ex 113 . . 3  |-  ( ph  ->  ( v  e.  (
ZZ>= `  C )  -> 
v  e.  dom  ran  R ) )
4847ssrdv 3016 . 2  |-  ( ph  ->  ( ZZ>= `  C )  C_ 
dom  ran  R )
4912, 48eqssd 3027 1  |-  ( ph  ->  dom  ran  R  =  ( ZZ>= `  C )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    = wceq 1285    e. wcel 1434    C_ wss 2984   <.cop 3425    |-> cmpt 3865   omcom 4368    X. cxp 4399   `'ccnv 4400   dom cdm 4401   ran crn 4402   Fun wfun 4963   -->wf 4965   -1-1-onto->wf1o 4968   ` cfv 4969  (class class class)co 5591    |-> cmpt2 5593   1stc1st 5844   2ndc2nd 5845  freccfrec 6087   1c1 7254    + caddc 7256   ZZcz 8646   ZZ>=cuz 8914
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-coll 3919  ax-sep 3922  ax-nul 3930  ax-pow 3974  ax-pr 4000  ax-un 4224  ax-setind 4316  ax-iinf 4366  ax-cnex 7339  ax-resscn 7340  ax-1cn 7341  ax-1re 7342  ax-icn 7343  ax-addcl 7344  ax-addrcl 7345  ax-mulcl 7346  ax-addcom 7348  ax-addass 7350  ax-distr 7352  ax-i2m1 7353  ax-0lt1 7354  ax-0id 7356  ax-rnegex 7357  ax-cnre 7359  ax-pre-ltirr 7360  ax-pre-ltwlin 7361  ax-pre-lttrn 7362  ax-pre-ltadd 7364
This theorem depends on definitions:  df-bi 115  df-3or 921  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ne 2250  df-nel 2345  df-ral 2358  df-rex 2359  df-reu 2360  df-rab 2362  df-v 2614  df-sbc 2827  df-csb 2920  df-dif 2986  df-un 2988  df-in 2990  df-ss 2997  df-nul 3270  df-pw 3408  df-sn 3428  df-pr 3429  df-op 3431  df-uni 3628  df-int 3663  df-iun 3706  df-br 3812  df-opab 3866  df-mpt 3867  df-tr 3902  df-id 4084  df-iord 4157  df-on 4159  df-ilim 4160  df-suc 4162  df-iom 4369  df-xp 4407  df-rel 4408  df-cnv 4409  df-co 4410  df-dm 4411  df-rn 4412  df-res 4413  df-ima 4414  df-iota 4934  df-fun 4971  df-fn 4972  df-f 4973  df-f1 4974  df-fo 4975  df-f1o 4976  df-fv 4977  df-riota 5547  df-ov 5594  df-oprab 5595  df-mpt2 5596  df-1st 5846  df-2nd 5847  df-recs 6002  df-frec 6088  df-pnf 7427  df-mnf 7428  df-xr 7429  df-ltxr 7430  df-le 7431  df-sub 7558  df-neg 7559  df-inn 8317  df-n0 8566  df-z 8647  df-uz 8915
This theorem is referenced by:  frecuzrdgdom  9714
  Copyright terms: Public domain W3C validator