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

Theorem frecuzrdgdomlem 10430
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 10428 . . . . 5  |-  ( ph  ->  R : om --> ( (
ZZ>= `  C )  X.  S ) )
7 frn 5386 . . . . 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 4838 . . . 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 5071 . . 3  |-  dom  (
( ZZ>= `  C )  X.  S )  C_  ( ZZ>=
`  C )
1210, 11sstrdi 3179 . 2  |-  ( ph  ->  dom  ran  R  C_  ( ZZ>=
`  C ) )
138adantr 276 . . . . . . . . 9  |-  ( (
ph  /\  v  e.  ( ZZ>= `  C )
)  ->  ran  R  C_  ( ( ZZ>= `  C
)  X.  S ) )
14 ffun 5380 . . . . . . . . . . . 12  |-  ( R : om --> ( (
ZZ>= `  C )  X.  S )  ->  Fun  R )
156, 14syl 14 . . . . . . . . . . 11  |-  ( ph  ->  Fun  R )
1615adantr 276 . . . . . . . . . 10  |-  ( (
ph  /\  v  e.  ( ZZ>= `  C )
)  ->  Fun  R )
17 frecuzrdgdomlem.g . . . . . . . . . . . . 13  |-  G  = frec ( ( x  e.  ZZ  |->  ( x  + 
1 ) ) ,  C )
181, 17frec2uzf1od 10419 . . . . . . . . . . . 12  |-  ( ph  ->  G : om -1-1-onto-> ( ZZ>= `  C )
)
19 f1ocnvdm 5795 . . . . . . . . . . . 12  |-  ( ( G : om -1-1-onto-> ( ZZ>= `  C )  /\  v  e.  ( ZZ>=
`  C ) )  ->  ( `' G `  v )  e.  om )
2018, 19sylan 283 . . . . . . . . . . 11  |-  ( (
ph  /\  v  e.  ( ZZ>= `  C )
)  ->  ( `' G `  v )  e.  om )
21 fdm 5383 . . . . . . . . . . . . 13  |-  ( R : om --> ( (
ZZ>= `  C )  X.  S )  ->  dom  R  =  om )
226, 21syl 14 . . . . . . . . . . . 12  |-  ( ph  ->  dom  R  =  om )
2322adantr 276 . . . . . . . . . . 11  |-  ( (
ph  /\  v  e.  ( ZZ>= `  C )
)  ->  dom  R  =  om )
2420, 23eleqtrrd 2267 . . . . . . . . . 10  |-  ( (
ph  /\  v  e.  ( ZZ>= `  C )
)  ->  ( `' G `  v )  e.  dom  R )
25 fvelrn 5660 . . . . . . . . . 10  |-  ( ( Fun  R  /\  ( `' G `  v )  e.  dom  R )  ->  ( R `  ( `' G `  v ) )  e.  ran  R
)
2616, 24, 25syl2anc 411 . . . . . . . . 9  |-  ( (
ph  /\  v  e.  ( ZZ>= `  C )
)  ->  ( R `  ( `' G `  v ) )  e. 
ran  R )
2713, 26sseldd 3168 . . . . . . . 8  |-  ( (
ph  /\  v  e.  ( ZZ>= `  C )
)  ->  ( R `  ( `' G `  v ) )  e.  ( ( ZZ>= `  C
)  X.  S ) )
28 1st2nd2 6189 . . . . . . . 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 276 . . . . . . . . . 10  |-  ( (
ph  /\  v  e.  ( ZZ>= `  C )
)  ->  C  e.  ZZ )
312adantr 276 . . . . . . . . . 10  |-  ( (
ph  /\  v  e.  ( ZZ>= `  C )
)  ->  A  e.  S )
323adantr 276 . . . . . . . . . 10  |-  ( (
ph  /\  v  e.  ( ZZ>= `  C )
)  ->  S  C_  T
)
334adantlr 477 . . . . . . . . . 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 10429 . . . . . . . . 9  |-  ( (
ph  /\  v  e.  ( ZZ>= `  C )
)  ->  ( 1st `  ( R `  ( `' G `  v ) ) )  =  ( G `  ( `' G `  v ) ) )
35 f1ocnvfv2 5792 . . . . . . . . . 10  |-  ( ( G : om -1-1-onto-> ( ZZ>= `  C )  /\  v  e.  ( ZZ>=
`  C ) )  ->  ( G `  ( `' G `  v ) )  =  v )
3618, 35sylan 283 . . . . . . . . 9  |-  ( (
ph  /\  v  e.  ( ZZ>= `  C )
)  ->  ( G `  ( `' G `  v ) )  =  v )
3734, 36eqtrd 2220 . . . . . . . 8  |-  ( (
ph  /\  v  e.  ( ZZ>= `  C )
)  ->  ( 1st `  ( R `  ( `' G `  v ) ) )  =  v )
3837opeq1d 3796 . . . . . . 7  |-  ( (
ph  /\  v  e.  ( ZZ>= `  C )
)  ->  <. ( 1st `  ( R `  ( `' G `  v ) ) ) ,  ( 2nd `  ( R `
 ( `' G `  v ) ) )
>.  =  <. v ,  ( 2nd `  ( R `  ( `' G `  v )
) ) >. )
3929, 38eqtrd 2220 . . . . . 6  |-  ( (
ph  /\  v  e.  ( ZZ>= `  C )
)  ->  ( R `  ( `' G `  v ) )  = 
<. v ,  ( 2nd `  ( R `  ( `' G `  v ) ) ) >. )
4039, 26eqeltrrd 2265 . . . . 5  |-  ( (
ph  /\  v  e.  ( ZZ>= `  C )
)  ->  <. v ,  ( 2nd `  ( R `  ( `' G `  v )
) ) >.  e.  ran  R )
41 simpr 110 . . . . . 6  |-  ( (
ph  /\  v  e.  ( ZZ>= `  C )
)  ->  v  e.  ( ZZ>= `  C )
)
42 xp2nd 6180 . . . . . . 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 4844 . . . . . 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 411 . . . . 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 115 . . 3  |-  ( ph  ->  ( v  e.  (
ZZ>= `  C )  -> 
v  e.  dom  ran  R ) )
4847ssrdv 3173 . 2  |-  ( ph  ->  ( ZZ>= `  C )  C_ 
dom  ran  R )
4912, 48eqssd 3184 1  |-  ( ph  ->  dom  ran  R  =  ( ZZ>= `  C )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    = wceq 1363    e. wcel 2158    C_ wss 3141   <.cop 3607    |-> cmpt 4076   omcom 4601    X. cxp 4636   `'ccnv 4637   dom cdm 4638   ran crn 4639   Fun wfun 5222   -->wf 5224   -1-1-onto->wf1o 5227   ` cfv 5228  (class class class)co 5888    e. cmpo 5890   1stc1st 6152   2ndc2nd 6153  freccfrec 6404   1c1 7825    + caddc 7827   ZZcz 9266   ZZ>=cuz 9541
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-13 2160  ax-14 2161  ax-ext 2169  ax-coll 4130  ax-sep 4133  ax-nul 4141  ax-pow 4186  ax-pr 4221  ax-un 4445  ax-setind 4548  ax-iinf 4599  ax-cnex 7915  ax-resscn 7916  ax-1cn 7917  ax-1re 7918  ax-icn 7919  ax-addcl 7920  ax-addrcl 7921  ax-mulcl 7922  ax-addcom 7924  ax-addass 7926  ax-distr 7928  ax-i2m1 7929  ax-0lt1 7930  ax-0id 7932  ax-rnegex 7933  ax-cnre 7935  ax-pre-ltirr 7936  ax-pre-ltwlin 7937  ax-pre-lttrn 7938  ax-pre-ltadd 7940
This theorem depends on definitions:  df-bi 117  df-3or 980  df-3an 981  df-tru 1366  df-fal 1369  df-nf 1471  df-sb 1773  df-eu 2039  df-mo 2040  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ne 2358  df-nel 2453  df-ral 2470  df-rex 2471  df-reu 2472  df-rab 2474  df-v 2751  df-sbc 2975  df-csb 3070  df-dif 3143  df-un 3145  df-in 3147  df-ss 3154  df-nul 3435  df-pw 3589  df-sn 3610  df-pr 3611  df-op 3613  df-uni 3822  df-int 3857  df-iun 3900  df-br 4016  df-opab 4077  df-mpt 4078  df-tr 4114  df-id 4305  df-iord 4378  df-on 4380  df-ilim 4381  df-suc 4383  df-iom 4602  df-xp 4644  df-rel 4645  df-cnv 4646  df-co 4647  df-dm 4648  df-rn 4649  df-res 4650  df-ima 4651  df-iota 5190  df-fun 5230  df-fn 5231  df-f 5232  df-f1 5233  df-fo 5234  df-f1o 5235  df-fv 5236  df-riota 5844  df-ov 5891  df-oprab 5892  df-mpo 5893  df-1st 6154  df-2nd 6155  df-recs 6319  df-frec 6405  df-pnf 8007  df-mnf 8008  df-xr 8009  df-ltxr 8010  df-le 8011  df-sub 8143  df-neg 8144  df-inn 8933  df-n0 9190  df-z 9267  df-uz 9542
This theorem is referenced by:  frecuzrdgdom  10431
  Copyright terms: Public domain W3C validator