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Theorem frecuzrdgrcl 10076
Description: The function  R (used in the definition of the recursive definition generator on upper integers) is a function defined for all natural numbers. (Contributed by Jim Kingdon, 1-Apr-2022.)
Hypotheses
Ref Expression
frec2uz.1  |-  ( ph  ->  C  e.  ZZ )
frec2uz.2  |-  G  = frec ( ( x  e.  ZZ  |->  ( x  + 
1 ) ) ,  C )
frecuzrdgrrn.a  |-  ( ph  ->  A  e.  S )
frecuzrdgrrn.f  |-  ( (
ph  /\  ( x  e.  ( ZZ>= `  C )  /\  y  e.  S
) )  ->  (
x F y )  e.  S )
frecuzrdgrrn.2  |-  R  = frec ( ( x  e.  ( ZZ>= `  C ) ,  y  e.  S  |-> 
<. ( x  +  1 ) ,  ( x F y ) >.
) ,  <. C ,  A >. )
Assertion
Ref Expression
frecuzrdgrcl  |-  ( ph  ->  R : om --> ( (
ZZ>= `  C )  X.  S ) )
Distinct variable groups:    y, A    x, C, y    y, G    x, F, y    x, S, y    ph, x, y
Allowed substitution hints:    A( x)    R( x, y)    G( x)

Proof of Theorem frecuzrdgrcl
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 1st2nd2 6027 . . . . . . 7  |-  ( z  e.  ( ( ZZ>= `  C )  X.  S
)  ->  z  =  <. ( 1st `  z
) ,  ( 2nd `  z ) >. )
21adantl 273 . . . . . 6  |-  ( (
ph  /\  z  e.  ( ( ZZ>= `  C
)  X.  S ) )  ->  z  =  <. ( 1st `  z
) ,  ( 2nd `  z ) >. )
32fveq2d 5379 . . . . 5  |-  ( (
ph  /\  z  e.  ( ( ZZ>= `  C
)  X.  S ) )  ->  ( (
x  e.  ( ZZ>= `  C ) ,  y  e.  S  |->  <. (
x  +  1 ) ,  ( x F y ) >. ) `  z )  =  ( ( x  e.  (
ZZ>= `  C ) ,  y  e.  S  |->  <.
( x  +  1 ) ,  ( x F y ) >.
) `  <. ( 1st `  z ) ,  ( 2nd `  z )
>. ) )
4 df-ov 5731 . . . . . . 7  |-  ( ( 1st `  z ) ( x  e.  (
ZZ>= `  C ) ,  y  e.  S  |->  <.
( x  +  1 ) ,  ( x F y ) >.
) ( 2nd `  z
) )  =  ( ( x  e.  (
ZZ>= `  C ) ,  y  e.  S  |->  <.
( x  +  1 ) ,  ( x F y ) >.
) `  <. ( 1st `  z ) ,  ( 2nd `  z )
>. )
5 xp1st 6017 . . . . . . . . 9  |-  ( z  e.  ( ( ZZ>= `  C )  X.  S
)  ->  ( 1st `  z )  e.  (
ZZ>= `  C ) )
65adantl 273 . . . . . . . 8  |-  ( (
ph  /\  z  e.  ( ( ZZ>= `  C
)  X.  S ) )  ->  ( 1st `  z )  e.  (
ZZ>= `  C ) )
7 xp2nd 6018 . . . . . . . . 9  |-  ( z  e.  ( ( ZZ>= `  C )  X.  S
)  ->  ( 2nd `  z )  e.  S
)
87adantl 273 . . . . . . . 8  |-  ( (
ph  /\  z  e.  ( ( ZZ>= `  C
)  X.  S ) )  ->  ( 2nd `  z )  e.  S
)
9 peano2uz 9280 . . . . . . . . . 10  |-  ( ( 1st `  z )  e.  ( ZZ>= `  C
)  ->  ( ( 1st `  z )  +  1 )  e.  (
ZZ>= `  C ) )
106, 9syl 14 . . . . . . . . 9  |-  ( (
ph  /\  z  e.  ( ( ZZ>= `  C
)  X.  S ) )  ->  ( ( 1st `  z )  +  1 )  e.  (
ZZ>= `  C ) )
11 frecuzrdgrrn.f . . . . . . . . . . . 12  |-  ( (
ph  /\  ( x  e.  ( ZZ>= `  C )  /\  y  e.  S
) )  ->  (
x F y )  e.  S )
1211ralrimivva 2488 . . . . . . . . . . 11  |-  ( ph  ->  A. x  e.  (
ZZ>= `  C ) A. y  e.  S  (
x F y )  e.  S )
1312adantr 272 . . . . . . . . . 10  |-  ( (
ph  /\  z  e.  ( ( ZZ>= `  C
)  X.  S ) )  ->  A. x  e.  ( ZZ>= `  C ) A. y  e.  S  ( x F y )  e.  S )
14 oveq1 5735 . . . . . . . . . . . . 13  |-  ( x  =  ( 1st `  z
)  ->  ( x F y )  =  ( ( 1st `  z
) F y ) )
1514eleq1d 2183 . . . . . . . . . . . 12  |-  ( x  =  ( 1st `  z
)  ->  ( (
x F y )  e.  S  <->  ( ( 1st `  z ) F y )  e.  S
) )
16 oveq2 5736 . . . . . . . . . . . . 13  |-  ( y  =  ( 2nd `  z
)  ->  ( ( 1st `  z ) F y )  =  ( ( 1st `  z
) F ( 2nd `  z ) ) )
1716eleq1d 2183 . . . . . . . . . . . 12  |-  ( y  =  ( 2nd `  z
)  ->  ( (
( 1st `  z
) F y )  e.  S  <->  ( ( 1st `  z ) F ( 2nd `  z
) )  e.  S
) )
1815, 17rspc2v 2772 . . . . . . . . . . 11  |-  ( ( ( 1st `  z
)  e.  ( ZZ>= `  C )  /\  ( 2nd `  z )  e.  S )  ->  ( A. x  e.  ( ZZ>=
`  C ) A. y  e.  S  (
x F y )  e.  S  ->  (
( 1st `  z
) F ( 2nd `  z ) )  e.  S ) )
196, 8, 18syl2anc 406 . . . . . . . . . 10  |-  ( (
ph  /\  z  e.  ( ( ZZ>= `  C
)  X.  S ) )  ->  ( A. x  e.  ( ZZ>= `  C ) A. y  e.  S  ( x F y )  e.  S  ->  ( ( 1st `  z ) F ( 2nd `  z
) )  e.  S
) )
2013, 19mpd 13 . . . . . . . . 9  |-  ( (
ph  /\  z  e.  ( ( ZZ>= `  C
)  X.  S ) )  ->  ( ( 1st `  z ) F ( 2nd `  z
) )  e.  S
)
21 opelxp 4529 . . . . . . . . 9  |-  ( <.
( ( 1st `  z
)  +  1 ) ,  ( ( 1st `  z ) F ( 2nd `  z ) ) >.  e.  (
( ZZ>= `  C )  X.  S )  <->  ( (
( 1st `  z
)  +  1 )  e.  ( ZZ>= `  C
)  /\  ( ( 1st `  z ) F ( 2nd `  z
) )  e.  S
) )
2210, 20, 21sylanbrc 411 . . . . . . . 8  |-  ( (
ph  /\  z  e.  ( ( ZZ>= `  C
)  X.  S ) )  ->  <. ( ( 1st `  z )  +  1 ) ,  ( ( 1st `  z
) F ( 2nd `  z ) ) >.  e.  ( ( ZZ>= `  C
)  X.  S ) )
23 oveq1 5735 . . . . . . . . . 10  |-  ( x  =  ( 1st `  z
)  ->  ( x  +  1 )  =  ( ( 1st `  z
)  +  1 ) )
2423, 14opeq12d 3679 . . . . . . . . 9  |-  ( x  =  ( 1st `  z
)  ->  <. ( x  +  1 ) ,  ( x F y ) >.  =  <. ( ( 1st `  z
)  +  1 ) ,  ( ( 1st `  z ) F y ) >. )
2516opeq2d 3678 . . . . . . . . 9  |-  ( y  =  ( 2nd `  z
)  ->  <. ( ( 1st `  z )  +  1 ) ,  ( ( 1st `  z
) F y )
>.  =  <. ( ( 1st `  z )  +  1 ) ,  ( ( 1st `  z
) F ( 2nd `  z ) ) >.
)
26 eqid 2115 . . . . . . . . 9  |-  ( x  e.  ( ZZ>= `  C
) ,  y  e.  S  |->  <. ( x  + 
1 ) ,  ( x F y )
>. )  =  (
x  e.  ( ZZ>= `  C ) ,  y  e.  S  |->  <. (
x  +  1 ) ,  ( x F y ) >. )
2724, 25, 26ovmpog 5859 . . . . . . . 8  |-  ( ( ( 1st `  z
)  e.  ( ZZ>= `  C )  /\  ( 2nd `  z )  e.  S  /\  <. (
( 1st `  z
)  +  1 ) ,  ( ( 1st `  z ) F ( 2nd `  z ) ) >.  e.  (
( ZZ>= `  C )  X.  S ) )  -> 
( ( 1st `  z
) ( x  e.  ( ZZ>= `  C ) ,  y  e.  S  |-> 
<. ( x  +  1 ) ,  ( x F y ) >.
) ( 2nd `  z
) )  =  <. ( ( 1st `  z
)  +  1 ) ,  ( ( 1st `  z ) F ( 2nd `  z ) ) >. )
286, 8, 22, 27syl3anc 1199 . . . . . . 7  |-  ( (
ph  /\  z  e.  ( ( ZZ>= `  C
)  X.  S ) )  ->  ( ( 1st `  z ) ( x  e.  ( ZZ>= `  C ) ,  y  e.  S  |->  <. (
x  +  1 ) ,  ( x F y ) >. )
( 2nd `  z
) )  =  <. ( ( 1st `  z
)  +  1 ) ,  ( ( 1st `  z ) F ( 2nd `  z ) ) >. )
294, 28syl5eqr 2161 . . . . . 6  |-  ( (
ph  /\  z  e.  ( ( ZZ>= `  C
)  X.  S ) )  ->  ( (
x  e.  ( ZZ>= `  C ) ,  y  e.  S  |->  <. (
x  +  1 ) ,  ( x F y ) >. ) `  <. ( 1st `  z
) ,  ( 2nd `  z ) >. )  =  <. ( ( 1st `  z )  +  1 ) ,  ( ( 1st `  z ) F ( 2nd `  z
) ) >. )
3029, 22eqeltrd 2191 . . . . 5  |-  ( (
ph  /\  z  e.  ( ( ZZ>= `  C
)  X.  S ) )  ->  ( (
x  e.  ( ZZ>= `  C ) ,  y  e.  S  |->  <. (
x  +  1 ) ,  ( x F y ) >. ) `  <. ( 1st `  z
) ,  ( 2nd `  z ) >. )  e.  ( ( ZZ>= `  C
)  X.  S ) )
313, 30eqeltrd 2191 . . . 4  |-  ( (
ph  /\  z  e.  ( ( ZZ>= `  C
)  X.  S ) )  ->  ( (
x  e.  ( ZZ>= `  C ) ,  y  e.  S  |->  <. (
x  +  1 ) ,  ( x F y ) >. ) `  z )  e.  ( ( ZZ>= `  C )  X.  S ) )
3231ralrimiva 2479 . . 3  |-  ( ph  ->  A. z  e.  ( ( ZZ>= `  C )  X.  S ) ( ( x  e.  ( ZZ>= `  C ) ,  y  e.  S  |->  <. (
x  +  1 ) ,  ( x F y ) >. ) `  z )  e.  ( ( ZZ>= `  C )  X.  S ) )
33 frec2uz.1 . . . . 5  |-  ( ph  ->  C  e.  ZZ )
34 uzid 9242 . . . . 5  |-  ( C  e.  ZZ  ->  C  e.  ( ZZ>= `  C )
)
3533, 34syl 14 . . . 4  |-  ( ph  ->  C  e.  ( ZZ>= `  C ) )
36 frecuzrdgrrn.a . . . 4  |-  ( ph  ->  A  e.  S )
37 opelxp 4529 . . . 4  |-  ( <. C ,  A >.  e.  ( ( ZZ>= `  C
)  X.  S )  <-> 
( C  e.  (
ZZ>= `  C )  /\  A  e.  S )
)
3835, 36, 37sylanbrc 411 . . 3  |-  ( ph  -> 
<. C ,  A >.  e.  ( ( ZZ>= `  C
)  X.  S ) )
39 frecfcl 6256 . . 3  |-  ( ( A. z  e.  ( ( ZZ>= `  C )  X.  S ) ( ( x  e.  ( ZZ>= `  C ) ,  y  e.  S  |->  <. (
x  +  1 ) ,  ( x F y ) >. ) `  z )  e.  ( ( ZZ>= `  C )  X.  S )  /\  <. C ,  A >.  e.  ( ( ZZ>= `  C )  X.  S ) )  -> frec ( ( x  e.  ( ZZ>= `  C ) ,  y  e.  S  |-> 
<. ( x  +  1 ) ,  ( x F y ) >.
) ,  <. C ,  A >. ) : om --> ( ( ZZ>= `  C
)  X.  S ) )
4032, 38, 39syl2anc 406 . 2  |-  ( ph  -> frec ( ( x  e.  ( ZZ>= `  C ) ,  y  e.  S  |-> 
<. ( x  +  1 ) ,  ( x F y ) >.
) ,  <. C ,  A >. ) : om --> ( ( ZZ>= `  C
)  X.  S ) )
41 frecuzrdgrrn.2 . . 3  |-  R  = frec ( ( x  e.  ( ZZ>= `  C ) ,  y  e.  S  |-> 
<. ( x  +  1 ) ,  ( x F y ) >.
) ,  <. C ,  A >. )
4241feq1i 5223 . 2  |-  ( R : om --> ( (
ZZ>= `  C )  X.  S )  <-> frec ( (
x  e.  ( ZZ>= `  C ) ,  y  e.  S  |->  <. (
x  +  1 ) ,  ( x F y ) >. ) ,  <. C ,  A >. ) : om --> ( (
ZZ>= `  C )  X.  S ) )
4340, 42sylibr 133 1  |-  ( ph  ->  R : om --> ( (
ZZ>= `  C )  X.  S ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    = wceq 1314    e. wcel 1463   A.wral 2390   <.cop 3496    |-> cmpt 3949   omcom 4464    X. cxp 4497   -->wf 5077   ` cfv 5081  (class class class)co 5728    e. cmpo 5730   1stc1st 5990   2ndc2nd 5991  freccfrec 6241   1c1 7548    + caddc 7550   ZZcz 8958   ZZ>=cuz 9228
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 586  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-13 1474  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-coll 4003  ax-sep 4006  ax-nul 4014  ax-pow 4058  ax-pr 4091  ax-un 4315  ax-setind 4412  ax-iinf 4462  ax-cnex 7636  ax-resscn 7637  ax-1cn 7638  ax-1re 7639  ax-icn 7640  ax-addcl 7641  ax-addrcl 7642  ax-mulcl 7643  ax-addcom 7645  ax-addass 7647  ax-distr 7649  ax-i2m1 7650  ax-0lt1 7651  ax-0id 7653  ax-rnegex 7654  ax-cnre 7656  ax-pre-ltirr 7657  ax-pre-ltwlin 7658  ax-pre-lttrn 7659  ax-pre-ltadd 7661
This theorem depends on definitions:  df-bi 116  df-3or 946  df-3an 947  df-tru 1317  df-fal 1320  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2244  df-ne 2283  df-nel 2378  df-ral 2395  df-rex 2396  df-reu 2397  df-rab 2399  df-v 2659  df-sbc 2879  df-csb 2972  df-dif 3039  df-un 3041  df-in 3043  df-ss 3050  df-nul 3330  df-pw 3478  df-sn 3499  df-pr 3500  df-op 3502  df-uni 3703  df-int 3738  df-iun 3781  df-br 3896  df-opab 3950  df-mpt 3951  df-tr 3987  df-id 4175  df-iord 4248  df-on 4250  df-ilim 4251  df-suc 4253  df-iom 4465  df-xp 4505  df-rel 4506  df-cnv 4507  df-co 4508  df-dm 4509  df-rn 4510  df-res 4511  df-ima 4512  df-iota 5046  df-fun 5083  df-fn 5084  df-f 5085  df-f1 5086  df-fo 5087  df-f1o 5088  df-fv 5089  df-riota 5684  df-ov 5731  df-oprab 5732  df-mpo 5733  df-1st 5992  df-2nd 5993  df-recs 6156  df-frec 6242  df-pnf 7726  df-mnf 7727  df-xr 7728  df-ltxr 7729  df-le 7730  df-sub 7858  df-neg 7859  df-inn 8631  df-n0 8882  df-z 8959  df-uz 9229
This theorem is referenced by:  frecuzrdglem  10077  frecuzrdgtcl  10078  frecuzrdg0  10079
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