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Theorem uzrdgfni 11253
Description: The recursive definition generator on upper integers is a function. See comment in om2uzrdg 11251. (Contributed by Mario Carneiro, 26-Jun-2013.) (Revised by Mario Carneiro, 4-May-2015.)
Hypotheses
Ref Expression
om2uz.1  |-  C  e.  ZZ
om2uz.2  |-  G  =  ( rec ( ( x  e.  _V  |->  ( x  +  1 ) ) ,  C )  |`  om )
uzrdg.1  |-  A  e. 
_V
uzrdg.2  |-  R  =  ( rec ( ( x  e.  _V , 
y  e.  _V  |->  <.
( x  +  1 ) ,  ( x F y ) >.
) ,  <. C ,  A >. )  |`  om )
uzrdg.3  |-  S  =  ran  R
Assertion
Ref Expression
uzrdgfni  |-  S  Fn  ( ZZ>= `  C )
Distinct variable groups:    y, A    x, y, C    y, G    x, F, y
Allowed substitution hints:    A( x)    R( x, y)    S( x, y)    G( x)

Proof of Theorem uzrdgfni
Dummy variables  z  w  v are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 uzrdg.3 . . . . . . . . 9  |-  S  =  ran  R
21eleq2i 2468 . . . . . . . 8  |-  ( z  e.  S  <->  z  e.  ran  R )
3 frfnom 6651 . . . . . . . . . 10  |-  ( rec ( ( x  e. 
_V ,  y  e. 
_V  |->  <. ( x  + 
1 ) ,  ( x F y )
>. ) ,  <. C ,  A >. )  |`  om )  Fn  om
4 uzrdg.2 . . . . . . . . . . 11  |-  R  =  ( rec ( ( x  e.  _V , 
y  e.  _V  |->  <.
( x  +  1 ) ,  ( x F y ) >.
) ,  <. C ,  A >. )  |`  om )
54fneq1i 5498 . . . . . . . . . 10  |-  ( R  Fn  om  <->  ( rec ( ( x  e. 
_V ,  y  e. 
_V  |->  <. ( x  + 
1 ) ,  ( x F y )
>. ) ,  <. C ,  A >. )  |`  om )  Fn  om )
63, 5mpbir 201 . . . . . . . . 9  |-  R  Fn  om
7 fvelrnb 5733 . . . . . . . . 9  |-  ( R  Fn  om  ->  (
z  e.  ran  R  <->  E. w  e.  om  ( R `  w )  =  z ) )
86, 7ax-mp 8 . . . . . . . 8  |-  ( z  e.  ran  R  <->  E. w  e.  om  ( R `  w )  =  z )
92, 8bitri 241 . . . . . . 7  |-  ( z  e.  S  <->  E. w  e.  om  ( R `  w )  =  z )
10 om2uz.1 . . . . . . . . . . 11  |-  C  e.  ZZ
11 om2uz.2 . . . . . . . . . . 11  |-  G  =  ( rec ( ( x  e.  _V  |->  ( x  +  1 ) ) ,  C )  |`  om )
12 uzrdg.1 . . . . . . . . . . 11  |-  A  e. 
_V
1310, 11, 12, 4om2uzrdg 11251 . . . . . . . . . 10  |-  ( w  e.  om  ->  ( R `  w )  =  <. ( G `  w ) ,  ( 2nd `  ( R `
 w ) )
>. )
1410, 11om2uzuzi 11244 . . . . . . . . . . 11  |-  ( w  e.  om  ->  ( G `  w )  e.  ( ZZ>= `  C )
)
15 fvex 5701 . . . . . . . . . . 11  |-  ( 2nd `  ( R `  w
) )  e.  _V
16 opelxpi 4869 . . . . . . . . . . 11  |-  ( ( ( G `  w
)  e.  ( ZZ>= `  C )  /\  ( 2nd `  ( R `  w ) )  e. 
_V )  ->  <. ( G `  w ) ,  ( 2nd `  ( R `  w )
) >.  e.  ( (
ZZ>= `  C )  X. 
_V ) )
1714, 15, 16sylancl 644 . . . . . . . . . 10  |-  ( w  e.  om  ->  <. ( G `  w ) ,  ( 2nd `  ( R `  w )
) >.  e.  ( (
ZZ>= `  C )  X. 
_V ) )
1813, 17eqeltrd 2478 . . . . . . . . 9  |-  ( w  e.  om  ->  ( R `  w )  e.  ( ( ZZ>= `  C
)  X.  _V )
)
19 eleq1 2464 . . . . . . . . 9  |-  ( ( R `  w )  =  z  ->  (
( R `  w
)  e.  ( (
ZZ>= `  C )  X. 
_V )  <->  z  e.  ( ( ZZ>= `  C
)  X.  _V )
) )
2018, 19syl5ibcom 212 . . . . . . . 8  |-  ( w  e.  om  ->  (
( R `  w
)  =  z  -> 
z  e.  ( (
ZZ>= `  C )  X. 
_V ) ) )
2120rexlimiv 2784 . . . . . . 7  |-  ( E. w  e.  om  ( R `  w )  =  z  ->  z  e.  ( ( ZZ>= `  C
)  X.  _V )
)
229, 21sylbi 188 . . . . . 6  |-  ( z  e.  S  ->  z  e.  ( ( ZZ>= `  C
)  X.  _V )
)
2322ssriv 3312 . . . . 5  |-  S  C_  ( ( ZZ>= `  C
)  X.  _V )
24 xpss 4941 . . . . 5  |-  ( (
ZZ>= `  C )  X. 
_V )  C_  ( _V  X.  _V )
2523, 24sstri 3317 . . . 4  |-  S  C_  ( _V  X.  _V )
26 df-rel 4844 . . . 4  |-  ( Rel 
S  <->  S  C_  ( _V 
X.  _V ) )
2725, 26mpbir 201 . . 3  |-  Rel  S
28 fvex 5701 . . . . . 6  |-  ( 2nd `  ( R `  ( `' G `  v ) ) )  e.  _V
29 eqeq2 2413 . . . . . . . 8  |-  ( w  =  ( 2nd `  ( R `  ( `' G `  v )
) )  ->  (
z  =  w  <->  z  =  ( 2nd `  ( R `
 ( `' G `  v ) ) ) ) )
3029imbi2d 308 . . . . . . 7  |-  ( w  =  ( 2nd `  ( R `  ( `' G `  v )
) )  ->  (
( <. v ,  z
>.  e.  S  ->  z  =  w )  <->  ( <. v ,  z >.  e.  S  ->  z  =  ( 2nd `  ( R `  ( `' G `  v ) ) ) ) ) )
3130albidv 1632 . . . . . 6  |-  ( w  =  ( 2nd `  ( R `  ( `' G `  v )
) )  ->  ( A. z ( <. v ,  z >.  e.  S  ->  z  =  w )  <->  A. z ( <. v ,  z >.  e.  S  ->  z  =  ( 2nd `  ( R `  ( `' G `  v ) ) ) ) ) )
3228, 31spcev 3003 . . . . 5  |-  ( A. z ( <. v ,  z >.  e.  S  ->  z  =  ( 2nd `  ( R `  ( `' G `  v ) ) ) )  ->  E. w A. z (
<. v ,  z >.  e.  S  ->  z  =  w ) )
331eleq2i 2468 . . . . . . 7  |-  ( <.
v ,  z >.  e.  S  <->  <. v ,  z
>.  e.  ran  R )
34 fvelrnb 5733 . . . . . . . 8  |-  ( R  Fn  om  ->  ( <. v ,  z >.  e.  ran  R  <->  E. w  e.  om  ( R `  w )  =  <. v ,  z >. )
)
356, 34ax-mp 8 . . . . . . 7  |-  ( <.
v ,  z >.  e.  ran  R  <->  E. w  e.  om  ( R `  w )  =  <. v ,  z >. )
3633, 35bitri 241 . . . . . 6  |-  ( <.
v ,  z >.  e.  S  <->  E. w  e.  om  ( R `  w )  =  <. v ,  z
>. )
3713eqeq1d 2412 . . . . . . . . . . . 12  |-  ( w  e.  om  ->  (
( R `  w
)  =  <. v ,  z >.  <->  <. ( G `
 w ) ,  ( 2nd `  ( R `  w )
) >.  =  <. v ,  z >. )
)
38 fvex 5701 . . . . . . . . . . . . 13  |-  ( G `
 w )  e. 
_V
3938, 15opth1 4394 . . . . . . . . . . . 12  |-  ( <.
( G `  w
) ,  ( 2nd `  ( R `  w
) ) >.  =  <. v ,  z >.  ->  ( G `  w )  =  v )
4037, 39syl6bi 220 . . . . . . . . . . 11  |-  ( w  e.  om  ->  (
( R `  w
)  =  <. v ,  z >.  ->  ( G `  w )  =  v ) )
4110, 11om2uzf1oi 11248 . . . . . . . . . . . 12  |-  G : om
-1-1-onto-> ( ZZ>= `  C )
42 f1ocnvfv 5975 . . . . . . . . . . . 12  |-  ( ( G : om -1-1-onto-> ( ZZ>= `  C )  /\  w  e.  om )  ->  ( ( G `
 w )  =  v  ->  ( `' G `  v )  =  w ) )
4341, 42mpan 652 . . . . . . . . . . 11  |-  ( w  e.  om  ->  (
( G `  w
)  =  v  -> 
( `' G `  v )  =  w ) )
4440, 43syld 42 . . . . . . . . . 10  |-  ( w  e.  om  ->  (
( R `  w
)  =  <. v ,  z >.  ->  ( `' G `  v )  =  w ) )
45 fveq2 5687 . . . . . . . . . . 11  |-  ( ( `' G `  v )  =  w  ->  ( R `  ( `' G `  v )
)  =  ( R `
 w ) )
4645fveq2d 5691 . . . . . . . . . 10  |-  ( ( `' G `  v )  =  w  ->  ( 2nd `  ( R `  ( `' G `  v ) ) )  =  ( 2nd `  ( R `
 w ) ) )
4744, 46syl6 31 . . . . . . . . 9  |-  ( w  e.  om  ->  (
( R `  w
)  =  <. v ,  z >.  ->  ( 2nd `  ( R `  ( `' G `  v ) ) )  =  ( 2nd `  ( R `
 w ) ) ) )
4847imp 419 . . . . . . . 8  |-  ( ( w  e.  om  /\  ( R `  w )  =  <. v ,  z
>. )  ->  ( 2nd `  ( R `  ( `' G `  v ) ) )  =  ( 2nd `  ( R `
 w ) ) )
49 vex 2919 . . . . . . . . . 10  |-  v  e. 
_V
50 vex 2919 . . . . . . . . . 10  |-  z  e. 
_V
5149, 50op2ndd 6317 . . . . . . . . 9  |-  ( ( R `  w )  =  <. v ,  z
>.  ->  ( 2nd `  ( R `  w )
)  =  z )
5251adantl 453 . . . . . . . 8  |-  ( ( w  e.  om  /\  ( R `  w )  =  <. v ,  z
>. )  ->  ( 2nd `  ( R `  w
) )  =  z )
5348, 52eqtr2d 2437 . . . . . . 7  |-  ( ( w  e.  om  /\  ( R `  w )  =  <. v ,  z
>. )  ->  z  =  ( 2nd `  ( R `  ( `' G `  v )
) ) )
5453rexlimiva 2785 . . . . . 6  |-  ( E. w  e.  om  ( R `  w )  =  <. v ,  z
>.  ->  z  =  ( 2nd `  ( R `
 ( `' G `  v ) ) ) )
5536, 54sylbi 188 . . . . 5  |-  ( <.
v ,  z >.  e.  S  ->  z  =  ( 2nd `  ( R `  ( `' G `  v )
) ) )
5632, 55mpg 1554 . . . 4  |-  E. w A. z ( <. v ,  z >.  e.  S  ->  z  =  w )
5756ax-gen 1552 . . 3  |-  A. v E. w A. z (
<. v ,  z >.  e.  S  ->  z  =  w )
58 dffun5 5426 . . 3  |-  ( Fun 
S  <->  ( Rel  S  /\  A. v E. w A. z ( <. v ,  z >.  e.  S  ->  z  =  w ) ) )
5927, 57, 58mpbir2an 887 . 2  |-  Fun  S
60 dmss 5028 . . . . 5  |-  ( S 
C_  ( ( ZZ>= `  C )  X.  _V )  ->  dom  S  C_  dom  ( ( ZZ>= `  C
)  X.  _V )
)
6123, 60ax-mp 8 . . . 4  |-  dom  S  C_ 
dom  ( ( ZZ>= `  C )  X.  _V )
62 dmxpss 5259 . . . 4  |-  dom  (
( ZZ>= `  C )  X.  _V )  C_  ( ZZ>=
`  C )
6361, 62sstri 3317 . . 3  |-  dom  S  C_  ( ZZ>= `  C )
6410, 11, 12, 4uzrdglem 11252 . . . . . 6  |-  ( v  e.  ( ZZ>= `  C
)  ->  <. v ,  ( 2nd `  ( R `  ( `' G `  v )
) ) >.  e.  ran  R )
6564, 1syl6eleqr 2495 . . . . 5  |-  ( v  e.  ( ZZ>= `  C
)  ->  <. v ,  ( 2nd `  ( R `  ( `' G `  v )
) ) >.  e.  S
)
6649, 28opeldm 5032 . . . . 5  |-  ( <.
v ,  ( 2nd `  ( R `  ( `' G `  v ) ) ) >.  e.  S  ->  v  e.  dom  S
)
6765, 66syl 16 . . . 4  |-  ( v  e.  ( ZZ>= `  C
)  ->  v  e.  dom  S )
6867ssriv 3312 . . 3  |-  ( ZZ>= `  C )  C_  dom  S
6963, 68eqssi 3324 . 2  |-  dom  S  =  ( ZZ>= `  C
)
70 df-fn 5416 . 2  |-  ( S  Fn  ( ZZ>= `  C
)  <->  ( Fun  S  /\  dom  S  =  (
ZZ>= `  C ) ) )
7159, 69, 70mpbir2an 887 1  |-  S  Fn  ( ZZ>= `  C )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359   A.wal 1546   E.wex 1547    = wceq 1649    e. wcel 1721   E.wrex 2667   _Vcvv 2916    C_ wss 3280   <.cop 3777    e. cmpt 4226   omcom 4804    X. cxp 4835   `'ccnv 4836   dom cdm 4837   ran crn 4838    |` cres 4839   Rel wrel 4842   Fun wfun 5407    Fn wfn 5408   -1-1-onto->wf1o 5412   ` cfv 5413  (class class class)co 6040    e. cmpt2 6042   2ndc2nd 6307   reccrdg 6626   1c1 8947    + caddc 8949   ZZcz 10238   ZZ>=cuz 10444
This theorem is referenced by:  uzrdg0i  11254  uzrdgsuci  11255  seqfn  11290
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-2nd 6309  df-riota 6508  df-recs 6592  df-rdg 6627  df-er 6864  df-en 7069  df-dom 7070  df-sdom 7071  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-nn 9957  df-n0 10178  df-z 10239  df-uz 10445
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