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Theorem frecuzrdgdomlem 9824
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 9822 . . . . 5  |-  ( ph  ->  R : om --> ( (
ZZ>= `  C )  X.  S ) )
7 frn 5169 . . . . 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 4635 . . . 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 4861 . . 3  |-  dom  (
( ZZ>= `  C )  X.  S )  C_  ( ZZ>=
`  C )
1210, 11syl6ss 3037 . 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 5164 . . . . . . . . . . . 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 9813 . . . . . . . . . . . 12  |-  ( ph  ->  G : om -1-1-onto-> ( ZZ>= `  C )
)
19 f1ocnvdm 5560 . . . . . . . . . . . 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 5166 . . . . . . . . . . . . 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 2167 . . . . . . . . . 10  |-  ( (
ph  /\  v  e.  ( ZZ>= `  C )
)  ->  ( `' G `  v )  e.  dom  R )
25 fvelrn 5430 . . . . . . . . . 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 3026 . . . . . . . 8  |-  ( (
ph  /\  v  e.  ( ZZ>= `  C )
)  ->  ( R `  ( `' G `  v ) )  e.  ( ( ZZ>= `  C
)  X.  S ) )
28 1st2nd2 5945 . . . . . . . 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 9823 . . . . . . . . 9  |-  ( (
ph  /\  v  e.  ( ZZ>= `  C )
)  ->  ( 1st `  ( R `  ( `' G `  v ) ) )  =  ( G `  ( `' G `  v ) ) )
35 f1ocnvfv2 5557 . . . . . . . . . 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 2120 . . . . . . . 8  |-  ( (
ph  /\  v  e.  ( ZZ>= `  C )
)  ->  ( 1st `  ( R `  ( `' G `  v ) ) )  =  v )
3837opeq1d 3628 . . . . . . 7  |-  ( (
ph  /\  v  e.  ( ZZ>= `  C )
)  ->  <. ( 1st `  ( R `  ( `' G `  v ) ) ) ,  ( 2nd `  ( R `
 ( `' G `  v ) ) )
>.  =  <. v ,  ( 2nd `  ( R `  ( `' G `  v )
) ) >. )
3929, 38eqtrd 2120 . . . . . 6  |-  ( (
ph  /\  v  e.  ( ZZ>= `  C )
)  ->  ( R `  ( `' G `  v ) )  = 
<. v ,  ( 2nd `  ( R `  ( `' G `  v ) ) ) >. )
4039, 26eqeltrrd 2165 . . . . 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 5937 . . . . . . 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 4641 . . . . . 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 3031 . 2  |-  ( ph  ->  ( ZZ>= `  C )  C_ 
dom  ran  R )
4912, 48eqssd 3042 1  |-  ( ph  ->  dom  ran  R  =  ( ZZ>= `  C )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    = wceq 1289    e. wcel 1438    C_ wss 2999   <.cop 3449    |-> cmpt 3899   omcom 4405    X. cxp 4436   `'ccnv 4437   dom cdm 4438   ran crn 4439   Fun wfun 5009   -->wf 5011   -1-1-onto->wf1o 5014   ` cfv 5015  (class class class)co 5652    |-> cmpt2 5654   1stc1st 5909   2ndc2nd 5910  freccfrec 6155   1c1 7351    + caddc 7353   ZZcz 8750   ZZ>=cuz 9019
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 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-coll 3954  ax-sep 3957  ax-nul 3965  ax-pow 4009  ax-pr 4036  ax-un 4260  ax-setind 4353  ax-iinf 4403  ax-cnex 7436  ax-resscn 7437  ax-1cn 7438  ax-1re 7439  ax-icn 7440  ax-addcl 7441  ax-addrcl 7442  ax-mulcl 7443  ax-addcom 7445  ax-addass 7447  ax-distr 7449  ax-i2m1 7450  ax-0lt1 7451  ax-0id 7453  ax-rnegex 7454  ax-cnre 7456  ax-pre-ltirr 7457  ax-pre-ltwlin 7458  ax-pre-lttrn 7459  ax-pre-ltadd 7461
This theorem depends on definitions:  df-bi 115  df-3or 925  df-3an 926  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ne 2256  df-nel 2351  df-ral 2364  df-rex 2365  df-reu 2366  df-rab 2368  df-v 2621  df-sbc 2841  df-csb 2934  df-dif 3001  df-un 3003  df-in 3005  df-ss 3012  df-nul 3287  df-pw 3431  df-sn 3452  df-pr 3453  df-op 3455  df-uni 3654  df-int 3689  df-iun 3732  df-br 3846  df-opab 3900  df-mpt 3901  df-tr 3937  df-id 4120  df-iord 4193  df-on 4195  df-ilim 4196  df-suc 4198  df-iom 4406  df-xp 4444  df-rel 4445  df-cnv 4446  df-co 4447  df-dm 4448  df-rn 4449  df-res 4450  df-ima 4451  df-iota 4980  df-fun 5017  df-fn 5018  df-f 5019  df-f1 5020  df-fo 5021  df-f1o 5022  df-fv 5023  df-riota 5608  df-ov 5655  df-oprab 5656  df-mpt2 5657  df-1st 5911  df-2nd 5912  df-recs 6070  df-frec 6156  df-pnf 7524  df-mnf 7525  df-xr 7526  df-ltxr 7527  df-le 7528  df-sub 7655  df-neg 7656  df-inn 8423  df-n0 8674  df-z 8751  df-uz 9020
This theorem is referenced by:  frecuzrdgdom  9825
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