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Theorem frecuzrdgdomlem 10083
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 10081 . . . . 5  |-  ( ph  ->  R : om --> ( (
ZZ>= `  C )  X.  S ) )
7 frn 5239 . . . . 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 4698 . . . 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 4927 . . 3  |-  dom  (
( ZZ>= `  C )  X.  S )  C_  ( ZZ>=
`  C )
1210, 11syl6ss 3075 . 2  |-  ( ph  ->  dom  ran  R  C_  ( ZZ>=
`  C ) )
138adantr 272 . . . . . . . . 9  |-  ( (
ph  /\  v  e.  ( ZZ>= `  C )
)  ->  ran  R  C_  ( ( ZZ>= `  C
)  X.  S ) )
14 ffun 5233 . . . . . . . . . . . 12  |-  ( R : om --> ( (
ZZ>= `  C )  X.  S )  ->  Fun  R )
156, 14syl 14 . . . . . . . . . . 11  |-  ( ph  ->  Fun  R )
1615adantr 272 . . . . . . . . . 10  |-  ( (
ph  /\  v  e.  ( ZZ>= `  C )
)  ->  Fun  R )
17 frecuzrdgdomlem.g . . . . . . . . . . . . 13  |-  G  = frec ( ( x  e.  ZZ  |->  ( x  + 
1 ) ) ,  C )
181, 17frec2uzf1od 10072 . . . . . . . . . . . 12  |-  ( ph  ->  G : om -1-1-onto-> ( ZZ>= `  C )
)
19 f1ocnvdm 5636 . . . . . . . . . . . 12  |-  ( ( G : om -1-1-onto-> ( ZZ>= `  C )  /\  v  e.  ( ZZ>=
`  C ) )  ->  ( `' G `  v )  e.  om )
2018, 19sylan 279 . . . . . . . . . . 11  |-  ( (
ph  /\  v  e.  ( ZZ>= `  C )
)  ->  ( `' G `  v )  e.  om )
21 fdm 5236 . . . . . . . . . . . . 13  |-  ( R : om --> ( (
ZZ>= `  C )  X.  S )  ->  dom  R  =  om )
226, 21syl 14 . . . . . . . . . . . 12  |-  ( ph  ->  dom  R  =  om )
2322adantr 272 . . . . . . . . . . 11  |-  ( (
ph  /\  v  e.  ( ZZ>= `  C )
)  ->  dom  R  =  om )
2420, 23eleqtrrd 2194 . . . . . . . . . 10  |-  ( (
ph  /\  v  e.  ( ZZ>= `  C )
)  ->  ( `' G `  v )  e.  dom  R )
25 fvelrn 5505 . . . . . . . . . 10  |-  ( ( Fun  R  /\  ( `' G `  v )  e.  dom  R )  ->  ( R `  ( `' G `  v ) )  e.  ran  R
)
2616, 24, 25syl2anc 406 . . . . . . . . 9  |-  ( (
ph  /\  v  e.  ( ZZ>= `  C )
)  ->  ( R `  ( `' G `  v ) )  e. 
ran  R )
2713, 26sseldd 3064 . . . . . . . 8  |-  ( (
ph  /\  v  e.  ( ZZ>= `  C )
)  ->  ( R `  ( `' G `  v ) )  e.  ( ( ZZ>= `  C
)  X.  S ) )
28 1st2nd2 6027 . . . . . . . 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 272 . . . . . . . . . 10  |-  ( (
ph  /\  v  e.  ( ZZ>= `  C )
)  ->  C  e.  ZZ )
312adantr 272 . . . . . . . . . 10  |-  ( (
ph  /\  v  e.  ( ZZ>= `  C )
)  ->  A  e.  S )
323adantr 272 . . . . . . . . . 10  |-  ( (
ph  /\  v  e.  ( ZZ>= `  C )
)  ->  S  C_  T
)
334adantlr 466 . . . . . . . . . 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 10082 . . . . . . . . 9  |-  ( (
ph  /\  v  e.  ( ZZ>= `  C )
)  ->  ( 1st `  ( R `  ( `' G `  v ) ) )  =  ( G `  ( `' G `  v ) ) )
35 f1ocnvfv2 5633 . . . . . . . . . 10  |-  ( ( G : om -1-1-onto-> ( ZZ>= `  C )  /\  v  e.  ( ZZ>=
`  C ) )  ->  ( G `  ( `' G `  v ) )  =  v )
3618, 35sylan 279 . . . . . . . . 9  |-  ( (
ph  /\  v  e.  ( ZZ>= `  C )
)  ->  ( G `  ( `' G `  v ) )  =  v )
3734, 36eqtrd 2147 . . . . . . . 8  |-  ( (
ph  /\  v  e.  ( ZZ>= `  C )
)  ->  ( 1st `  ( R `  ( `' G `  v ) ) )  =  v )
3837opeq1d 3677 . . . . . . 7  |-  ( (
ph  /\  v  e.  ( ZZ>= `  C )
)  ->  <. ( 1st `  ( R `  ( `' G `  v ) ) ) ,  ( 2nd `  ( R `
 ( `' G `  v ) ) )
>.  =  <. v ,  ( 2nd `  ( R `  ( `' G `  v )
) ) >. )
3929, 38eqtrd 2147 . . . . . 6  |-  ( (
ph  /\  v  e.  ( ZZ>= `  C )
)  ->  ( R `  ( `' G `  v ) )  = 
<. v ,  ( 2nd `  ( R `  ( `' G `  v ) ) ) >. )
4039, 26eqeltrrd 2192 . . . . 5  |-  ( (
ph  /\  v  e.  ( ZZ>= `  C )
)  ->  <. v ,  ( 2nd `  ( R `  ( `' G `  v )
) ) >.  e.  ran  R )
41 simpr 109 . . . . . 6  |-  ( (
ph  /\  v  e.  ( ZZ>= `  C )
)  ->  v  e.  ( ZZ>= `  C )
)
42 xp2nd 6018 . . . . . . 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 4704 . . . . . 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 406 . . . . 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 114 . . 3  |-  ( ph  ->  ( v  e.  (
ZZ>= `  C )  -> 
v  e.  dom  ran  R ) )
4847ssrdv 3069 . 2  |-  ( ph  ->  ( ZZ>= `  C )  C_ 
dom  ran  R )
4912, 48eqssd 3080 1  |-  ( ph  ->  dom  ran  R  =  ( ZZ>= `  C )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    = wceq 1314    e. wcel 1463    C_ wss 3037   <.cop 3496    |-> cmpt 3949   omcom 4464    X. cxp 4497   `'ccnv 4498   dom cdm 4499   ran crn 4500   Fun wfun 5075   -->wf 5077   -1-1-onto->wf1o 5080   ` cfv 5081  (class class class)co 5728    e. cmpo 5730   1stc1st 5990   2ndc2nd 5991  freccfrec 6241   1c1 7548    + caddc 7550   ZZcz 8958   ZZ>=cuz 9228
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 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-13 1474  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-coll 4003  ax-sep 4006  ax-nul 4014  ax-pow 4058  ax-pr 4091  ax-un 4315  ax-setind 4412  ax-iinf 4462  ax-cnex 7636  ax-resscn 7637  ax-1cn 7638  ax-1re 7639  ax-icn 7640  ax-addcl 7641  ax-addrcl 7642  ax-mulcl 7643  ax-addcom 7645  ax-addass 7647  ax-distr 7649  ax-i2m1 7650  ax-0lt1 7651  ax-0id 7653  ax-rnegex 7654  ax-cnre 7656  ax-pre-ltirr 7657  ax-pre-ltwlin 7658  ax-pre-lttrn 7659  ax-pre-ltadd 7661
This theorem depends on definitions:  df-bi 116  df-3or 946  df-3an 947  df-tru 1317  df-fal 1320  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2244  df-ne 2283  df-nel 2378  df-ral 2395  df-rex 2396  df-reu 2397  df-rab 2399  df-v 2659  df-sbc 2879  df-csb 2972  df-dif 3039  df-un 3041  df-in 3043  df-ss 3050  df-nul 3330  df-pw 3478  df-sn 3499  df-pr 3500  df-op 3502  df-uni 3703  df-int 3738  df-iun 3781  df-br 3896  df-opab 3950  df-mpt 3951  df-tr 3987  df-id 4175  df-iord 4248  df-on 4250  df-ilim 4251  df-suc 4253  df-iom 4465  df-xp 4505  df-rel 4506  df-cnv 4507  df-co 4508  df-dm 4509  df-rn 4510  df-res 4511  df-ima 4512  df-iota 5046  df-fun 5083  df-fn 5084  df-f 5085  df-f1 5086  df-fo 5087  df-f1o 5088  df-fv 5089  df-riota 5684  df-ov 5731  df-oprab 5732  df-mpo 5733  df-1st 5992  df-2nd 5993  df-recs 6156  df-frec 6242  df-pnf 7726  df-mnf 7727  df-xr 7728  df-ltxr 7729  df-le 7730  df-sub 7858  df-neg 7859  df-inn 8631  df-n0 8882  df-z 8959  df-uz 9229
This theorem is referenced by:  frecuzrdgdom  10084
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