MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  om2uzrani Structured version   Unicode version

Theorem om2uzrani 11285
Description: Range of  G (see om2uz0i 11280). (Contributed by NM, 3-Oct-2004.) (Revised by Mario Carneiro, 13-Sep-2013.)
Hypotheses
Ref Expression
om2uz.1  |-  C  e.  ZZ
om2uz.2  |-  G  =  ( rec ( ( x  e.  _V  |->  ( x  +  1 ) ) ,  C )  |`  om )
Assertion
Ref Expression
om2uzrani  |-  ran  G  =  ( ZZ>= `  C
)
Distinct variable group:    x, C
Allowed substitution hint:    G( x)

Proof of Theorem om2uzrani
Dummy variables  y 
z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 frfnom 6685 . . . . . 6  |-  ( rec ( ( x  e. 
_V  |->  ( x  + 
1 ) ) ,  C )  |`  om )  Fn  om
2 om2uz.2 . . . . . . 7  |-  G  =  ( rec ( ( x  e.  _V  |->  ( x  +  1 ) ) ,  C )  |`  om )
32fneq1i 5532 . . . . . 6  |-  ( G  Fn  om  <->  ( rec ( ( x  e. 
_V  |->  ( x  + 
1 ) ) ,  C )  |`  om )  Fn  om )
41, 3mpbir 201 . . . . 5  |-  G  Fn  om
5 fvelrnb 5767 . . . . 5  |-  ( G  Fn  om  ->  (
y  e.  ran  G  <->  E. z  e.  om  ( G `  z )  =  y ) )
64, 5ax-mp 8 . . . 4  |-  ( y  e.  ran  G  <->  E. z  e.  om  ( G `  z )  =  y )
7 om2uz.1 . . . . . . 7  |-  C  e.  ZZ
87, 2om2uzuzi 11282 . . . . . 6  |-  ( z  e.  om  ->  ( G `  z )  e.  ( ZZ>= `  C )
)
9 eleq1 2496 . . . . . 6  |-  ( ( G `  z )  =  y  ->  (
( G `  z
)  e.  ( ZZ>= `  C )  <->  y  e.  ( ZZ>= `  C )
) )
108, 9syl5ibcom 212 . . . . 5  |-  ( z  e.  om  ->  (
( G `  z
)  =  y  -> 
y  e.  ( ZZ>= `  C ) ) )
1110rexlimiv 2817 . . . 4  |-  ( E. z  e.  om  ( G `  z )  =  y  ->  y  e.  ( ZZ>= `  C )
)
126, 11sylbi 188 . . 3  |-  ( y  e.  ran  G  -> 
y  e.  ( ZZ>= `  C ) )
13 eleq1 2496 . . . 4  |-  ( z  =  C  ->  (
z  e.  ran  G  <->  C  e.  ran  G ) )
14 eleq1 2496 . . . 4  |-  ( z  =  y  ->  (
z  e.  ran  G  <->  y  e.  ran  G ) )
15 eleq1 2496 . . . 4  |-  ( z  =  ( y  +  1 )  ->  (
z  e.  ran  G  <->  ( y  +  1 )  e.  ran  G ) )
167, 2om2uz0i 11280 . . . . 5  |-  ( G `
 (/) )  =  C
17 peano1 4857 . . . . . 6  |-  (/)  e.  om
18 fnfvelrn 5860 . . . . . 6  |-  ( ( G  Fn  om  /\  (/) 
e.  om )  ->  ( G `  (/) )  e. 
ran  G )
194, 17, 18mp2an 654 . . . . 5  |-  ( G `
 (/) )  e.  ran  G
2016, 19eqeltrri 2507 . . . 4  |-  C  e. 
ran  G
217, 2om2uzsuci 11281 . . . . . . . . 9  |-  ( z  e.  om  ->  ( G `  suc  z )  =  ( ( G `
 z )  +  1 ) )
22 oveq1 6081 . . . . . . . . 9  |-  ( ( G `  z )  =  y  ->  (
( G `  z
)  +  1 )  =  ( y  +  1 ) )
2321, 22sylan9eq 2488 . . . . . . . 8  |-  ( ( z  e.  om  /\  ( G `  z )  =  y )  -> 
( G `  suc  z )  =  ( y  +  1 ) )
24 peano2 4858 . . . . . . . . . 10  |-  ( z  e.  om  ->  suc  z  e.  om )
25 fnfvelrn 5860 . . . . . . . . . 10  |-  ( ( G  Fn  om  /\  suc  z  e.  om )  ->  ( G `  suc  z )  e.  ran  G )
264, 24, 25sylancr 645 . . . . . . . . 9  |-  ( z  e.  om  ->  ( G `  suc  z )  e.  ran  G )
2726adantr 452 . . . . . . . 8  |-  ( ( z  e.  om  /\  ( G `  z )  =  y )  -> 
( G `  suc  z )  e.  ran  G )
2823, 27eqeltrrd 2511 . . . . . . 7  |-  ( ( z  e.  om  /\  ( G `  z )  =  y )  -> 
( y  +  1 )  e.  ran  G
)
2928rexlimiva 2818 . . . . . 6  |-  ( E. z  e.  om  ( G `  z )  =  y  ->  ( y  +  1 )  e. 
ran  G )
306, 29sylbi 188 . . . . 5  |-  ( y  e.  ran  G  -> 
( y  +  1 )  e.  ran  G
)
3130a1i 11 . . . 4  |-  ( y  e.  ( ZZ>= `  C
)  ->  ( y  e.  ran  G  ->  (
y  +  1 )  e.  ran  G ) )
327, 13, 14, 15, 14, 20, 31uzind4i 10531 . . 3  |-  ( y  e.  ( ZZ>= `  C
)  ->  y  e.  ran  G )
3312, 32impbii 181 . 2  |-  ( y  e.  ran  G  <->  y  e.  ( ZZ>= `  C )
)
3433eqriv 2433 1  |-  ran  G  =  ( ZZ>= `  C
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1652    e. wcel 1725   E.wrex 2699   _Vcvv 2949   (/)c0 3621    e. cmpt 4259   suc csuc 4576   omcom 4838   ran crn 4872    |` cres 4873    Fn wfn 5442   ` cfv 5447  (class class class)co 6074   reccrdg 6660   1c1 8984    + caddc 8986   ZZcz 10275   ZZ>=cuz 10481
This theorem is referenced by:  om2uzf1oi  11286
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4323  ax-nul 4331  ax-pow 4370  ax-pr 4396  ax-un 4694  ax-cnex 9039  ax-resscn 9040  ax-1cn 9041  ax-icn 9042  ax-addcl 9043  ax-addrcl 9044  ax-mulcl 9045  ax-mulrcl 9046  ax-mulcom 9047  ax-addass 9048  ax-mulass 9049  ax-distr 9050  ax-i2m1 9051  ax-1ne0 9052  ax-1rid 9053  ax-rnegex 9054  ax-rrecex 9055  ax-cnre 9056  ax-pre-lttri 9057  ax-pre-lttrn 9058  ax-pre-ltadd 9059  ax-pre-mulgt0 9060
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2703  df-rex 2704  df-reu 2705  df-rab 2707  df-v 2951  df-sbc 3155  df-csb 3245  df-dif 3316  df-un 3318  df-in 3320  df-ss 3327  df-pss 3329  df-nul 3622  df-if 3733  df-pw 3794  df-sn 3813  df-pr 3814  df-tp 3815  df-op 3816  df-uni 4009  df-iun 4088  df-br 4206  df-opab 4260  df-mpt 4261  df-tr 4296  df-eprel 4487  df-id 4491  df-po 4496  df-so 4497  df-fr 4534  df-we 4536  df-ord 4577  df-on 4578  df-lim 4579  df-suc 4580  df-om 4839  df-xp 4877  df-rel 4878  df-cnv 4879  df-co 4880  df-dm 4881  df-rn 4882  df-res 4883  df-ima 4884  df-iota 5411  df-fun 5449  df-fn 5450  df-f 5451  df-f1 5452  df-fo 5453  df-f1o 5454  df-fv 5455  df-ov 6077  df-oprab 6078  df-mpt2 6079  df-riota 6542  df-recs 6626  df-rdg 6661  df-er 6898  df-en 7103  df-dom 7104  df-sdom 7105  df-pnf 9115  df-mnf 9116  df-xr 9117  df-ltxr 9118  df-le 9119  df-sub 9286  df-neg 9287  df-nn 9994  df-n0 10215  df-z 10276  df-uz 10482
  Copyright terms: Public domain W3C validator