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Theorem om2uzrani 11031
Description: Range of  G (see om2uz0i 11026). (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 6463 . . . . . 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 5354 . . . . . 6  |-  ( G  Fn  om  <->  ( rec ( ( x  e. 
_V  |->  ( x  + 
1 ) ) ,  C )  |`  om )  Fn  om )
41, 3mpbir 200 . . . . 5  |-  G  Fn  om
5 fvelrnb 5586 . . . . 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 11028 . . . . . 6  |-  ( z  e.  om  ->  ( G `  z )  e.  ( ZZ>= `  C )
)
9 eleq1 2356 . . . . . 6  |-  ( ( G `  z )  =  y  ->  (
( G `  z
)  e.  ( ZZ>= `  C )  <->  y  e.  ( ZZ>= `  C )
) )
108, 9syl5ibcom 211 . . . . 5  |-  ( z  e.  om  ->  (
( G `  z
)  =  y  -> 
y  e.  ( ZZ>= `  C ) ) )
1110rexlimiv 2674 . . . 4  |-  ( E. z  e.  om  ( G `  z )  =  y  ->  y  e.  ( ZZ>= `  C )
)
126, 11sylbi 187 . . 3  |-  ( y  e.  ran  G  -> 
y  e.  ( ZZ>= `  C ) )
13 eleq1 2356 . . . 4  |-  ( z  =  C  ->  (
z  e.  ran  G  <->  C  e.  ran  G ) )
14 eleq1 2356 . . . 4  |-  ( z  =  y  ->  (
z  e.  ran  G  <->  y  e.  ran  G ) )
15 eleq1 2356 . . . 4  |-  ( z  =  ( y  +  1 )  ->  (
z  e.  ran  G  <->  ( y  +  1 )  e.  ran  G ) )
167, 2om2uz0i 11026 . . . . 5  |-  ( G `
 (/) )  =  C
17 peano1 4691 . . . . . 6  |-  (/)  e.  om
18 fnfvelrn 5678 . . . . . 6  |-  ( ( G  Fn  om  /\  (/) 
e.  om )  ->  ( G `  (/) )  e. 
ran  G )
194, 17, 18mp2an 653 . . . . 5  |-  ( G `
 (/) )  e.  ran  G
2016, 19eqeltrri 2367 . . . 4  |-  C  e. 
ran  G
217, 2om2uzsuci 11027 . . . . . . . . 9  |-  ( z  e.  om  ->  ( G `  suc  z )  =  ( ( G `
 z )  +  1 ) )
22 oveq1 5881 . . . . . . . . 9  |-  ( ( G `  z )  =  y  ->  (
( G `  z
)  +  1 )  =  ( y  +  1 ) )
2321, 22sylan9eq 2348 . . . . . . . 8  |-  ( ( z  e.  om  /\  ( G `  z )  =  y )  -> 
( G `  suc  z )  =  ( y  +  1 ) )
24 peano2 4692 . . . . . . . . . 10  |-  ( z  e.  om  ->  suc  z  e.  om )
25 fnfvelrn 5678 . . . . . . . . . 10  |-  ( ( G  Fn  om  /\  suc  z  e.  om )  ->  ( G `  suc  z )  e.  ran  G )
264, 24, 25sylancr 644 . . . . . . . . 9  |-  ( z  e.  om  ->  ( G `  suc  z )  e.  ran  G )
2726adantr 451 . . . . . . . 8  |-  ( ( z  e.  om  /\  ( G `  z )  =  y )  -> 
( G `  suc  z )  e.  ran  G )
2823, 27eqeltrrd 2371 . . . . . . 7  |-  ( ( z  e.  om  /\  ( G `  z )  =  y )  -> 
( y  +  1 )  e.  ran  G
)
2928rexlimiva 2675 . . . . . 6  |-  ( E. z  e.  om  ( G `  z )  =  y  ->  ( y  +  1 )  e. 
ran  G )
306, 29sylbi 187 . . . . 5  |-  ( y  e.  ran  G  -> 
( y  +  1 )  e.  ran  G
)
3130a1i 10 . . . 4  |-  ( y  e.  ( ZZ>= `  C
)  ->  ( y  e.  ran  G  ->  (
y  +  1 )  e.  ran  G ) )
327, 13, 14, 15, 14, 20, 31uzind4i 10296 . . 3  |-  ( y  e.  ( ZZ>= `  C
)  ->  y  e.  ran  G )
3312, 32impbii 180 . 2  |-  ( y  e.  ran  G  <->  y  e.  ( ZZ>= `  C )
)
3433eqriv 2293 1  |-  ran  G  =  ( ZZ>= `  C
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1632    e. wcel 1696   E.wrex 2557   _Vcvv 2801   (/)c0 3468    e. cmpt 4093   suc csuc 4410   omcom 4672   ran crn 4706    |` cres 4707    Fn wfn 5266   ` cfv 5271  (class class class)co 5874   reccrdg 6438   1c1 8754    + caddc 8756   ZZcz 10040   ZZ>=cuz 10246
This theorem is referenced by:  om2uzf1oi  11032
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-riota 6320  df-recs 6404  df-rdg 6439  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-nn 9763  df-n0 9982  df-z 10041  df-uz 10247
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