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Theorem frecuzrdgrcl 10138
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 6041 . . . . . . 7  |-  ( z  e.  ( ( ZZ>= `  C )  X.  S
)  ->  z  =  <. ( 1st `  z
) ,  ( 2nd `  z ) >. )
21adantl 275 . . . . . 6  |-  ( (
ph  /\  z  e.  ( ( ZZ>= `  C
)  X.  S ) )  ->  z  =  <. ( 1st `  z
) ,  ( 2nd `  z ) >. )
32fveq2d 5393 . . . . 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 5745 . . . . . . 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 6031 . . . . . . . . 9  |-  ( z  e.  ( ( ZZ>= `  C )  X.  S
)  ->  ( 1st `  z )  e.  (
ZZ>= `  C ) )
65adantl 275 . . . . . . . 8  |-  ( (
ph  /\  z  e.  ( ( ZZ>= `  C
)  X.  S ) )  ->  ( 1st `  z )  e.  (
ZZ>= `  C ) )
7 xp2nd 6032 . . . . . . . . 9  |-  ( z  e.  ( ( ZZ>= `  C )  X.  S
)  ->  ( 2nd `  z )  e.  S
)
87adantl 275 . . . . . . . 8  |-  ( (
ph  /\  z  e.  ( ( ZZ>= `  C
)  X.  S ) )  ->  ( 2nd `  z )  e.  S
)
9 peano2uz 9334 . . . . . . . . . 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 2491 . . . . . . . . . . 11  |-  ( ph  ->  A. x  e.  (
ZZ>= `  C ) A. y  e.  S  (
x F y )  e.  S )
1312adantr 274 . . . . . . . . . 10  |-  ( (
ph  /\  z  e.  ( ( ZZ>= `  C
)  X.  S ) )  ->  A. x  e.  ( ZZ>= `  C ) A. y  e.  S  ( x F y )  e.  S )
14 oveq1 5749 . . . . . . . . . . . . 13  |-  ( x  =  ( 1st `  z
)  ->  ( x F y )  =  ( ( 1st `  z
) F y ) )
1514eleq1d 2186 . . . . . . . . . . . 12  |-  ( x  =  ( 1st `  z
)  ->  ( (
x F y )  e.  S  <->  ( ( 1st `  z ) F y )  e.  S
) )
16 oveq2 5750 . . . . . . . . . . . . 13  |-  ( y  =  ( 2nd `  z
)  ->  ( ( 1st `  z ) F y )  =  ( ( 1st `  z
) F ( 2nd `  z ) ) )
1716eleq1d 2186 . . . . . . . . . . . 12  |-  ( y  =  ( 2nd `  z
)  ->  ( (
( 1st `  z
) F y )  e.  S  <->  ( ( 1st `  z ) F ( 2nd `  z
) )  e.  S
) )
1815, 17rspc2v 2776 . . . . . . . . . . 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 408 . . . . . . . . . 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 4539 . . . . . . . . 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 413 . . . . . . . 8  |-  ( (
ph  /\  z  e.  ( ( ZZ>= `  C
)  X.  S ) )  ->  <. ( ( 1st `  z )  +  1 ) ,  ( ( 1st `  z
) F ( 2nd `  z ) ) >.  e.  ( ( ZZ>= `  C
)  X.  S ) )
23 oveq1 5749 . . . . . . . . . 10  |-  ( x  =  ( 1st `  z
)  ->  ( x  +  1 )  =  ( ( 1st `  z
)  +  1 ) )
2423, 14opeq12d 3683 . . . . . . . . 9  |-  ( x  =  ( 1st `  z
)  ->  <. ( x  +  1 ) ,  ( x F y ) >.  =  <. ( ( 1st `  z
)  +  1 ) ,  ( ( 1st `  z ) F y ) >. )
2516opeq2d 3682 . . . . . . . . 9  |-  ( y  =  ( 2nd `  z
)  ->  <. ( ( 1st `  z )  +  1 ) ,  ( ( 1st `  z
) F y )
>.  =  <. ( ( 1st `  z )  +  1 ) ,  ( ( 1st `  z
) F ( 2nd `  z ) ) >.
)
26 eqid 2117 . . . . . . . . 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 5873 . . . . . . . 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 1201 . . . . . . 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 2164 . . . . . 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 2194 . . . . 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 2194 . . . 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 2482 . . 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 9296 . . . . 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 4539 . . . 4  |-  ( <. C ,  A >.  e.  ( ( ZZ>= `  C
)  X.  S )  <-> 
( C  e.  (
ZZ>= `  C )  /\  A  e.  S )
)
3835, 36, 37sylanbrc 413 . . 3  |-  ( ph  -> 
<. C ,  A >.  e.  ( ( ZZ>= `  C
)  X.  S ) )
39 frecfcl 6270 . . 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 408 . 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 5235 . 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 1316    e. wcel 1465   A.wral 2393   <.cop 3500    |-> cmpt 3959   omcom 4474    X. cxp 4507   -->wf 5089   ` cfv 5093  (class class class)co 5742    e. cmpo 5744   1stc1st 6004   2ndc2nd 6005  freccfrec 6255   1c1 7589    + caddc 7591   ZZcz 9012   ZZ>=cuz 9282
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 588  ax-in2 589  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-13 1476  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-coll 4013  ax-sep 4016  ax-nul 4024  ax-pow 4068  ax-pr 4101  ax-un 4325  ax-setind 4422  ax-iinf 4472  ax-cnex 7679  ax-resscn 7680  ax-1cn 7681  ax-1re 7682  ax-icn 7683  ax-addcl 7684  ax-addrcl 7685  ax-mulcl 7686  ax-addcom 7688  ax-addass 7690  ax-distr 7692  ax-i2m1 7693  ax-0lt1 7694  ax-0id 7696  ax-rnegex 7697  ax-cnre 7699  ax-pre-ltirr 7700  ax-pre-ltwlin 7701  ax-pre-lttrn 7702  ax-pre-ltadd 7704
This theorem depends on definitions:  df-bi 116  df-3or 948  df-3an 949  df-tru 1319  df-fal 1322  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ne 2286  df-nel 2381  df-ral 2398  df-rex 2399  df-reu 2400  df-rab 2402  df-v 2662  df-sbc 2883  df-csb 2976  df-dif 3043  df-un 3045  df-in 3047  df-ss 3054  df-nul 3334  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-uni 3707  df-int 3742  df-iun 3785  df-br 3900  df-opab 3960  df-mpt 3961  df-tr 3997  df-id 4185  df-iord 4258  df-on 4260  df-ilim 4261  df-suc 4263  df-iom 4475  df-xp 4515  df-rel 4516  df-cnv 4517  df-co 4518  df-dm 4519  df-rn 4520  df-res 4521  df-ima 4522  df-iota 5058  df-fun 5095  df-fn 5096  df-f 5097  df-f1 5098  df-fo 5099  df-f1o 5100  df-fv 5101  df-riota 5698  df-ov 5745  df-oprab 5746  df-mpo 5747  df-1st 6006  df-2nd 6007  df-recs 6170  df-frec 6256  df-pnf 7770  df-mnf 7771  df-xr 7772  df-ltxr 7773  df-le 7774  df-sub 7903  df-neg 7904  df-inn 8685  df-n0 8936  df-z 9013  df-uz 9283
This theorem is referenced by:  frecuzrdglem  10139  frecuzrdgtcl  10140  frecuzrdg0  10141
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