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Theorem frecuzrdgdomlem 10158
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 10156 . . . . 5  |-  ( ph  ->  R : om --> ( (
ZZ>= `  C )  X.  S ) )
7 frn 5251 . . . . 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 4708 . . . 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 4939 . . 3  |-  dom  (
( ZZ>= `  C )  X.  S )  C_  ( ZZ>=
`  C )
1210, 11sstrdi 3079 . 2  |-  ( ph  ->  dom  ran  R  C_  ( ZZ>=
`  C ) )
138adantr 274 . . . . . . . . 9  |-  ( (
ph  /\  v  e.  ( ZZ>= `  C )
)  ->  ran  R  C_  ( ( ZZ>= `  C
)  X.  S ) )
14 ffun 5245 . . . . . . . . . . . 12  |-  ( R : om --> ( (
ZZ>= `  C )  X.  S )  ->  Fun  R )
156, 14syl 14 . . . . . . . . . . 11  |-  ( ph  ->  Fun  R )
1615adantr 274 . . . . . . . . . 10  |-  ( (
ph  /\  v  e.  ( ZZ>= `  C )
)  ->  Fun  R )
17 frecuzrdgdomlem.g . . . . . . . . . . . . 13  |-  G  = frec ( ( x  e.  ZZ  |->  ( x  + 
1 ) ) ,  C )
181, 17frec2uzf1od 10147 . . . . . . . . . . . 12  |-  ( ph  ->  G : om -1-1-onto-> ( ZZ>= `  C )
)
19 f1ocnvdm 5650 . . . . . . . . . . . 12  |-  ( ( G : om -1-1-onto-> ( ZZ>= `  C )  /\  v  e.  ( ZZ>=
`  C ) )  ->  ( `' G `  v )  e.  om )
2018, 19sylan 281 . . . . . . . . . . 11  |-  ( (
ph  /\  v  e.  ( ZZ>= `  C )
)  ->  ( `' G `  v )  e.  om )
21 fdm 5248 . . . . . . . . . . . . 13  |-  ( R : om --> ( (
ZZ>= `  C )  X.  S )  ->  dom  R  =  om )
226, 21syl 14 . . . . . . . . . . . 12  |-  ( ph  ->  dom  R  =  om )
2322adantr 274 . . . . . . . . . . 11  |-  ( (
ph  /\  v  e.  ( ZZ>= `  C )
)  ->  dom  R  =  om )
2420, 23eleqtrrd 2197 . . . . . . . . . 10  |-  ( (
ph  /\  v  e.  ( ZZ>= `  C )
)  ->  ( `' G `  v )  e.  dom  R )
25 fvelrn 5519 . . . . . . . . . 10  |-  ( ( Fun  R  /\  ( `' G `  v )  e.  dom  R )  ->  ( R `  ( `' G `  v ) )  e.  ran  R
)
2616, 24, 25syl2anc 408 . . . . . . . . 9  |-  ( (
ph  /\  v  e.  ( ZZ>= `  C )
)  ->  ( R `  ( `' G `  v ) )  e. 
ran  R )
2713, 26sseldd 3068 . . . . . . . 8  |-  ( (
ph  /\  v  e.  ( ZZ>= `  C )
)  ->  ( R `  ( `' G `  v ) )  e.  ( ( ZZ>= `  C
)  X.  S ) )
28 1st2nd2 6041 . . . . . . . 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 274 . . . . . . . . . 10  |-  ( (
ph  /\  v  e.  ( ZZ>= `  C )
)  ->  C  e.  ZZ )
312adantr 274 . . . . . . . . . 10  |-  ( (
ph  /\  v  e.  ( ZZ>= `  C )
)  ->  A  e.  S )
323adantr 274 . . . . . . . . . 10  |-  ( (
ph  /\  v  e.  ( ZZ>= `  C )
)  ->  S  C_  T
)
334adantlr 468 . . . . . . . . . 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 10157 . . . . . . . . 9  |-  ( (
ph  /\  v  e.  ( ZZ>= `  C )
)  ->  ( 1st `  ( R `  ( `' G `  v ) ) )  =  ( G `  ( `' G `  v ) ) )
35 f1ocnvfv2 5647 . . . . . . . . . 10  |-  ( ( G : om -1-1-onto-> ( ZZ>= `  C )  /\  v  e.  ( ZZ>=
`  C ) )  ->  ( G `  ( `' G `  v ) )  =  v )
3618, 35sylan 281 . . . . . . . . 9  |-  ( (
ph  /\  v  e.  ( ZZ>= `  C )
)  ->  ( G `  ( `' G `  v ) )  =  v )
3734, 36eqtrd 2150 . . . . . . . 8  |-  ( (
ph  /\  v  e.  ( ZZ>= `  C )
)  ->  ( 1st `  ( R `  ( `' G `  v ) ) )  =  v )
3837opeq1d 3681 . . . . . . 7  |-  ( (
ph  /\  v  e.  ( ZZ>= `  C )
)  ->  <. ( 1st `  ( R `  ( `' G `  v ) ) ) ,  ( 2nd `  ( R `
 ( `' G `  v ) ) )
>.  =  <. v ,  ( 2nd `  ( R `  ( `' G `  v )
) ) >. )
3929, 38eqtrd 2150 . . . . . 6  |-  ( (
ph  /\  v  e.  ( ZZ>= `  C )
)  ->  ( R `  ( `' G `  v ) )  = 
<. v ,  ( 2nd `  ( R `  ( `' G `  v ) ) ) >. )
4039, 26eqeltrrd 2195 . . . . 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 6032 . . . . . . 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 4714 . . . . . 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 408 . . . . 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 3073 . 2  |-  ( ph  ->  ( ZZ>= `  C )  C_ 
dom  ran  R )
4912, 48eqssd 3084 1  |-  ( ph  ->  dom  ran  R  =  ( ZZ>= `  C )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    = wceq 1316    e. wcel 1465    C_ wss 3041   <.cop 3500    |-> cmpt 3959   omcom 4474    X. cxp 4507   `'ccnv 4508   dom cdm 4509   ran crn 4510   Fun wfun 5087   -->wf 5089   -1-1-onto->wf1o 5092   ` cfv 5093  (class class class)co 5742    e. cmpo 5744   1stc1st 6004   2ndc2nd 6005  freccfrec 6255   1c1 7589    + caddc 7591   ZZcz 9022   ZZ>=cuz 9294
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 588  ax-in2 589  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-13 1476  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-coll 4013  ax-sep 4016  ax-nul 4024  ax-pow 4068  ax-pr 4101  ax-un 4325  ax-setind 4422  ax-iinf 4472  ax-cnex 7679  ax-resscn 7680  ax-1cn 7681  ax-1re 7682  ax-icn 7683  ax-addcl 7684  ax-addrcl 7685  ax-mulcl 7686  ax-addcom 7688  ax-addass 7690  ax-distr 7692  ax-i2m1 7693  ax-0lt1 7694  ax-0id 7696  ax-rnegex 7697  ax-cnre 7699  ax-pre-ltirr 7700  ax-pre-ltwlin 7701  ax-pre-lttrn 7702  ax-pre-ltadd 7704
This theorem depends on definitions:  df-bi 116  df-3or 948  df-3an 949  df-tru 1319  df-fal 1322  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ne 2286  df-nel 2381  df-ral 2398  df-rex 2399  df-reu 2400  df-rab 2402  df-v 2662  df-sbc 2883  df-csb 2976  df-dif 3043  df-un 3045  df-in 3047  df-ss 3054  df-nul 3334  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-uni 3707  df-int 3742  df-iun 3785  df-br 3900  df-opab 3960  df-mpt 3961  df-tr 3997  df-id 4185  df-iord 4258  df-on 4260  df-ilim 4261  df-suc 4263  df-iom 4475  df-xp 4515  df-rel 4516  df-cnv 4517  df-co 4518  df-dm 4519  df-rn 4520  df-res 4521  df-ima 4522  df-iota 5058  df-fun 5095  df-fn 5096  df-f 5097  df-f1 5098  df-fo 5099  df-f1o 5100  df-fv 5101  df-riota 5698  df-ov 5745  df-oprab 5746  df-mpo 5747  df-1st 6006  df-2nd 6007  df-recs 6170  df-frec 6256  df-pnf 7770  df-mnf 7771  df-xr 7772  df-ltxr 7773  df-le 7774  df-sub 7903  df-neg 7904  df-inn 8689  df-n0 8946  df-z 9023  df-uz 9295
This theorem is referenced by:  frecuzrdgdom  10159
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