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Theorem frecuzrdgsuctlem 10409
Description: Successor value of a recursive definition generator on upper integers. See comment in frec2uz0d 10385 for the description of  G as the mapping from  om to  ( ZZ>= `  C ). (Contributed by Jim Kingdon, 29-Apr-2022.)
Hypotheses
Ref Expression
frecuzrdgrclt.c  |-  ( ph  ->  C  e.  ZZ )
frecuzrdgrclt.a  |-  ( ph  ->  A  e.  S )
frecuzrdgrclt.t  |-  ( ph  ->  S  C_  T )
frecuzrdgrclt.f  |-  ( (
ph  /\  ( x  e.  ( ZZ>= `  C )  /\  y  e.  S
) )  ->  (
x F y )  e.  S )
frecuzrdgrclt.r  |-  R  = frec ( ( x  e.  ( ZZ>= `  C ) ,  y  e.  T  |-> 
<. ( x  +  1 ) ,  ( x F y ) >.
) ,  <. C ,  A >. )
frecuzrdgsuctlem.g  |-  G  = frec ( ( x  e.  ZZ  |->  ( x  + 
1 ) ) ,  C )
frecuzrdgsuctlem.ran  |-  ( ph  ->  P  =  ran  R
)
Assertion
Ref Expression
frecuzrdgsuctlem  |-  ( (
ph  /\  B  e.  ( ZZ>= `  C )
)  ->  ( P `  ( B  +  1 ) )  =  ( B F ( P `
 B ) ) )
Distinct variable groups:    x, C, y   
x, F, y    x, S, y    x, T, y    ph, x, y    x, B, y    x, G, y   
x, R, y
Allowed substitution hints:    A( x, y)    P( x, y)

Proof of Theorem frecuzrdgsuctlem
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 frecuzrdgrclt.c . . . . . 6  |-  ( ph  ->  C  e.  ZZ )
2 frecuzrdgrclt.a . . . . . 6  |-  ( ph  ->  A  e.  S )
3 frecuzrdgrclt.t . . . . . 6  |-  ( ph  ->  S  C_  T )
4 frecuzrdgrclt.f . . . . . 6  |-  ( (
ph  /\  ( x  e.  ( ZZ>= `  C )  /\  y  e.  S
) )  ->  (
x F y )  e.  S )
5 frecuzrdgrclt.r . . . . . 6  |-  R  = frec ( ( x  e.  ( ZZ>= `  C ) ,  y  e.  T  |-> 
<. ( x  +  1 ) ,  ( x F y ) >.
) ,  <. C ,  A >. )
6 frecuzrdgsuctlem.ran . . . . . 6  |-  ( ph  ->  P  =  ran  R
)
71, 2, 3, 4, 5, 6frecuzrdgtclt 10407 . . . . 5  |-  ( ph  ->  P : ( ZZ>= `  C ) --> S )
87adantr 276 . . . 4  |-  ( (
ph  /\  B  e.  ( ZZ>= `  C )
)  ->  P :
( ZZ>= `  C ) --> S )
9 ffun 5364 . . . 4  |-  ( P : ( ZZ>= `  C
) --> S  ->  Fun  P )
108, 9syl 14 . . 3  |-  ( (
ph  /\  B  e.  ( ZZ>= `  C )
)  ->  Fun  P )
11 1st2nd2 6170 . . . . . . . . . . . . . . 15  |-  ( z  e.  ( ( ZZ>= `  C )  X.  S
)  ->  z  =  <. ( 1st `  z
) ,  ( 2nd `  z ) >. )
1211adantl 277 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  B  e.  ( ZZ>= `  C )
)  /\  z  e.  ( ( ZZ>= `  C
)  X.  S ) )  ->  z  =  <. ( 1st `  z
) ,  ( 2nd `  z ) >. )
1312fveq2d 5515 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  B  e.  ( ZZ>= `  C )
)  /\  z  e.  ( ( ZZ>= `  C
)  X.  S ) )  ->  ( (
x  e.  ( ZZ>= `  C ) ,  y  e.  T  |->  <. (
x  +  1 ) ,  ( x F y ) >. ) `  z )  =  ( ( x  e.  (
ZZ>= `  C ) ,  y  e.  T  |->  <.
( x  +  1 ) ,  ( x F y ) >.
) `  <. ( 1st `  z ) ,  ( 2nd `  z )
>. ) )
14 df-ov 5872 . . . . . . . . . . . . 13  |-  ( ( 1st `  z ) ( x  e.  (
ZZ>= `  C ) ,  y  e.  T  |->  <.
( x  +  1 ) ,  ( x F y ) >.
) ( 2nd `  z
) )  =  ( ( x  e.  (
ZZ>= `  C ) ,  y  e.  T  |->  <.
( x  +  1 ) ,  ( x F y ) >.
) `  <. ( 1st `  z ) ,  ( 2nd `  z )
>. )
1513, 14eqtr4di 2228 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  B  e.  ( ZZ>= `  C )
)  /\  z  e.  ( ( ZZ>= `  C
)  X.  S ) )  ->  ( (
x  e.  ( ZZ>= `  C ) ,  y  e.  T  |->  <. (
x  +  1 ) ,  ( x F y ) >. ) `  z )  =  ( ( 1st `  z
) ( x  e.  ( ZZ>= `  C ) ,  y  e.  T  |-> 
<. ( x  +  1 ) ,  ( x F y ) >.
) ( 2nd `  z
) ) )
16 xp1st 6160 . . . . . . . . . . . . . 14  |-  ( z  e.  ( ( ZZ>= `  C )  X.  S
)  ->  ( 1st `  z )  e.  (
ZZ>= `  C ) )
1716adantl 277 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  B  e.  ( ZZ>= `  C )
)  /\  z  e.  ( ( ZZ>= `  C
)  X.  S ) )  ->  ( 1st `  z )  e.  (
ZZ>= `  C ) )
183ad2antrr 488 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  B  e.  ( ZZ>= `  C )
)  /\  z  e.  ( ( ZZ>= `  C
)  X.  S ) )  ->  S  C_  T
)
19 xp2nd 6161 . . . . . . . . . . . . . . 15  |-  ( z  e.  ( ( ZZ>= `  C )  X.  S
)  ->  ( 2nd `  z )  e.  S
)
2019adantl 277 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  B  e.  ( ZZ>= `  C )
)  /\  z  e.  ( ( ZZ>= `  C
)  X.  S ) )  ->  ( 2nd `  z )  e.  S
)
2118, 20sseldd 3156 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  B  e.  ( ZZ>= `  C )
)  /\  z  e.  ( ( ZZ>= `  C
)  X.  S ) )  ->  ( 2nd `  z )  e.  T
)
22 peano2uz 9572 . . . . . . . . . . . . . . 15  |-  ( ( 1st `  z )  e.  ( ZZ>= `  C
)  ->  ( ( 1st `  z )  +  1 )  e.  (
ZZ>= `  C ) )
2317, 22syl 14 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  B  e.  ( ZZ>= `  C )
)  /\  z  e.  ( ( ZZ>= `  C
)  X.  S ) )  ->  ( ( 1st `  z )  +  1 )  e.  (
ZZ>= `  C ) )
24 oveq2 5877 . . . . . . . . . . . . . . . 16  |-  ( y  =  ( 2nd `  z
)  ->  ( ( 1st `  z ) F y )  =  ( ( 1st `  z
) F ( 2nd `  z ) ) )
2524eleq1d 2246 . . . . . . . . . . . . . . 15  |-  ( y  =  ( 2nd `  z
)  ->  ( (
( 1st `  z
) F y )  e.  S  <->  ( ( 1st `  z ) F ( 2nd `  z
) )  e.  S
) )
26 oveq1 5876 . . . . . . . . . . . . . . . . . 18  |-  ( x  =  ( 1st `  z
)  ->  ( x F y )  =  ( ( 1st `  z
) F y ) )
2726eleq1d 2246 . . . . . . . . . . . . . . . . 17  |-  ( x  =  ( 1st `  z
)  ->  ( (
x F y )  e.  S  <->  ( ( 1st `  z ) F y )  e.  S
) )
2827ralbidv 2477 . . . . . . . . . . . . . . . 16  |-  ( x  =  ( 1st `  z
)  ->  ( A. y  e.  S  (
x F y )  e.  S  <->  A. y  e.  S  ( ( 1st `  z ) F y )  e.  S
) )
294ralrimivva 2559 . . . . . . . . . . . . . . . . 17  |-  ( ph  ->  A. x  e.  (
ZZ>= `  C ) A. y  e.  S  (
x F y )  e.  S )
3029ad2antrr 488 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  B  e.  ( ZZ>= `  C )
)  /\  z  e.  ( ( ZZ>= `  C
)  X.  S ) )  ->  A. x  e.  ( ZZ>= `  C ) A. y  e.  S  ( x F y )  e.  S )
3128, 30, 17rspcdva 2846 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  B  e.  ( ZZ>= `  C )
)  /\  z  e.  ( ( ZZ>= `  C
)  X.  S ) )  ->  A. y  e.  S  ( ( 1st `  z ) F y )  e.  S
)
3225, 31, 20rspcdva 2846 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  B  e.  ( ZZ>= `  C )
)  /\  z  e.  ( ( ZZ>= `  C
)  X.  S ) )  ->  ( ( 1st `  z ) F ( 2nd `  z
) )  e.  S
)
33 opelxpi 4655 . . . . . . . . . . . . . 14  |-  ( ( ( ( 1st `  z
)  +  1 )  e.  ( ZZ>= `  C
)  /\  ( ( 1st `  z ) F ( 2nd `  z
) )  e.  S
)  ->  <. ( ( 1st `  z )  +  1 ) ,  ( ( 1st `  z
) F ( 2nd `  z ) ) >.  e.  ( ( ZZ>= `  C
)  X.  S ) )
3423, 32, 33syl2anc 411 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  B  e.  ( ZZ>= `  C )
)  /\  z  e.  ( ( ZZ>= `  C
)  X.  S ) )  ->  <. ( ( 1st `  z )  +  1 ) ,  ( ( 1st `  z
) F ( 2nd `  z ) ) >.  e.  ( ( ZZ>= `  C
)  X.  S ) )
35 oveq1 5876 . . . . . . . . . . . . . . 15  |-  ( x  =  ( 1st `  z
)  ->  ( x  +  1 )  =  ( ( 1st `  z
)  +  1 ) )
3635, 26opeq12d 3784 . . . . . . . . . . . . . 14  |-  ( x  =  ( 1st `  z
)  ->  <. ( x  +  1 ) ,  ( x F y ) >.  =  <. ( ( 1st `  z
)  +  1 ) ,  ( ( 1st `  z ) F y ) >. )
3724opeq2d 3783 . . . . . . . . . . . . . 14  |-  ( y  =  ( 2nd `  z
)  ->  <. ( ( 1st `  z )  +  1 ) ,  ( ( 1st `  z
) F y )
>.  =  <. ( ( 1st `  z )  +  1 ) ,  ( ( 1st `  z
) F ( 2nd `  z ) ) >.
)
38 eqid 2177 . . . . . . . . . . . . . 14  |-  ( x  e.  ( ZZ>= `  C
) ,  y  e.  T  |->  <. ( x  + 
1 ) ,  ( x F y )
>. )  =  (
x  e.  ( ZZ>= `  C ) ,  y  e.  T  |->  <. (
x  +  1 ) ,  ( x F y ) >. )
3936, 37, 38ovmpog 6003 . . . . . . . . . . . . 13  |-  ( ( ( 1st `  z
)  e.  ( ZZ>= `  C )  /\  ( 2nd `  z )  e.  T  /\  <. (
( 1st `  z
)  +  1 ) ,  ( ( 1st `  z ) F ( 2nd `  z ) ) >.  e.  (
( ZZ>= `  C )  X.  S ) )  -> 
( ( 1st `  z
) ( x  e.  ( ZZ>= `  C ) ,  y  e.  T  |-> 
<. ( x  +  1 ) ,  ( x F y ) >.
) ( 2nd `  z
) )  =  <. ( ( 1st `  z
)  +  1 ) ,  ( ( 1st `  z ) F ( 2nd `  z ) ) >. )
4017, 21, 34, 39syl3anc 1238 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  B  e.  ( ZZ>= `  C )
)  /\  z  e.  ( ( ZZ>= `  C
)  X.  S ) )  ->  ( ( 1st `  z ) ( x  e.  ( ZZ>= `  C ) ,  y  e.  T  |->  <. (
x  +  1 ) ,  ( x F y ) >. )
( 2nd `  z
) )  =  <. ( ( 1st `  z
)  +  1 ) ,  ( ( 1st `  z ) F ( 2nd `  z ) ) >. )
4115, 40eqtrd 2210 . . . . . . . . . . 11  |-  ( ( ( ph  /\  B  e.  ( ZZ>= `  C )
)  /\  z  e.  ( ( ZZ>= `  C
)  X.  S ) )  ->  ( (
x  e.  ( ZZ>= `  C ) ,  y  e.  T  |->  <. (
x  +  1 ) ,  ( x F y ) >. ) `  z )  =  <. ( ( 1st `  z
)  +  1 ) ,  ( ( 1st `  z ) F ( 2nd `  z ) ) >. )
4241, 34eqeltrd 2254 . . . . . . . . . 10  |-  ( ( ( ph  /\  B  e.  ( ZZ>= `  C )
)  /\  z  e.  ( ( ZZ>= `  C
)  X.  S ) )  ->  ( (
x  e.  ( ZZ>= `  C ) ,  y  e.  T  |->  <. (
x  +  1 ) ,  ( x F y ) >. ) `  z )  e.  ( ( ZZ>= `  C )  X.  S ) )
4342ralrimiva 2550 . . . . . . . . 9  |-  ( (
ph  /\  B  e.  ( ZZ>= `  C )
)  ->  A. z  e.  ( ( ZZ>= `  C
)  X.  S ) ( ( x  e.  ( ZZ>= `  C ) ,  y  e.  T  |-> 
<. ( x  +  1 ) ,  ( x F y ) >.
) `  z )  e.  ( ( ZZ>= `  C
)  X.  S ) )
44 uzid 9531 . . . . . . . . . . . 12  |-  ( C  e.  ZZ  ->  C  e.  ( ZZ>= `  C )
)
451, 44syl 14 . . . . . . . . . . 11  |-  ( ph  ->  C  e.  ( ZZ>= `  C ) )
46 opelxpi 4655 . . . . . . . . . . 11  |-  ( ( C  e.  ( ZZ>= `  C )  /\  A  e.  S )  ->  <. C ,  A >.  e.  ( (
ZZ>= `  C )  X.  S ) )
4745, 2, 46syl2anc 411 . . . . . . . . . 10  |-  ( ph  -> 
<. C ,  A >.  e.  ( ( ZZ>= `  C
)  X.  S ) )
4847adantr 276 . . . . . . . . 9  |-  ( (
ph  /\  B  e.  ( ZZ>= `  C )
)  ->  <. C ,  A >.  e.  ( (
ZZ>= `  C )  X.  S ) )
49 frecuzrdgsuctlem.g . . . . . . . . . . 11  |-  G  = frec ( ( x  e.  ZZ  |->  ( x  + 
1 ) ) ,  C )
501, 49frec2uzf1od 10392 . . . . . . . . . 10  |-  ( ph  ->  G : om -1-1-onto-> ( ZZ>= `  C )
)
51 f1ocnvdm 5776 . . . . . . . . . 10  |-  ( ( G : om -1-1-onto-> ( ZZ>= `  C )  /\  B  e.  ( ZZ>=
`  C ) )  ->  ( `' G `  B )  e.  om )
5250, 51sylan 283 . . . . . . . . 9  |-  ( (
ph  /\  B  e.  ( ZZ>= `  C )
)  ->  ( `' G `  B )  e.  om )
53 frecsuc 6402 . . . . . . . . 9  |-  ( ( A. z  e.  ( ( ZZ>= `  C )  X.  S ) ( ( x  e.  ( ZZ>= `  C ) ,  y  e.  T  |->  <. (
x  +  1 ) ,  ( x F y ) >. ) `  z )  e.  ( ( ZZ>= `  C )  X.  S )  /\  <. C ,  A >.  e.  ( ( ZZ>= `  C )  X.  S )  /\  ( `' G `  B )  e.  om )  -> 
(frec ( ( x  e.  ( ZZ>= `  C
) ,  y  e.  T  |->  <. ( x  + 
1 ) ,  ( x F y )
>. ) ,  <. C ,  A >. ) `  suc  ( `' G `  B ) )  =  ( ( x  e.  ( ZZ>= `  C ) ,  y  e.  T  |->  <. (
x  +  1 ) ,  ( x F y ) >. ) `  (frec ( ( x  e.  ( ZZ>= `  C
) ,  y  e.  T  |->  <. ( x  + 
1 ) ,  ( x F y )
>. ) ,  <. C ,  A >. ) `  ( `' G `  B ) ) ) )
5443, 48, 52, 53syl3anc 1238 . . . . . . . 8  |-  ( (
ph  /\  B  e.  ( ZZ>= `  C )
)  ->  (frec (
( x  e.  (
ZZ>= `  C ) ,  y  e.  T  |->  <.
( x  +  1 ) ,  ( x F y ) >.
) ,  <. C ,  A >. ) `  suc  ( `' G `  B ) )  =  ( ( x  e.  ( ZZ>= `  C ) ,  y  e.  T  |->  <. (
x  +  1 ) ,  ( x F y ) >. ) `  (frec ( ( x  e.  ( ZZ>= `  C
) ,  y  e.  T  |->  <. ( x  + 
1 ) ,  ( x F y )
>. ) ,  <. C ,  A >. ) `  ( `' G `  B ) ) ) )
555fveq1i 5512 . . . . . . . 8  |-  ( R `
 suc  ( `' G `  B )
)  =  (frec ( ( x  e.  (
ZZ>= `  C ) ,  y  e.  T  |->  <.
( x  +  1 ) ,  ( x F y ) >.
) ,  <. C ,  A >. ) `  suc  ( `' G `  B ) )
565fveq1i 5512 . . . . . . . . 9  |-  ( R `
 ( `' G `  B ) )  =  (frec ( ( x  e.  ( ZZ>= `  C
) ,  y  e.  T  |->  <. ( x  + 
1 ) ,  ( x F y )
>. ) ,  <. C ,  A >. ) `  ( `' G `  B ) )
5756fveq2i 5514 . . . . . . . 8  |-  ( ( x  e.  ( ZZ>= `  C ) ,  y  e.  T  |->  <. (
x  +  1 ) ,  ( x F y ) >. ) `  ( R `  ( `' G `  B ) ) )  =  ( ( x  e.  (
ZZ>= `  C ) ,  y  e.  T  |->  <.
( x  +  1 ) ,  ( x F y ) >.
) `  (frec (
( x  e.  (
ZZ>= `  C ) ,  y  e.  T  |->  <.
( x  +  1 ) ,  ( x F y ) >.
) ,  <. C ,  A >. ) `  ( `' G `  B ) ) )
5854, 55, 573eqtr4g 2235 . . . . . . 7  |-  ( (
ph  /\  B  e.  ( ZZ>= `  C )
)  ->  ( R `  suc  ( `' G `  B ) )  =  ( ( x  e.  ( ZZ>= `  C ) ,  y  e.  T  |-> 
<. ( x  +  1 ) ,  ( x F y ) >.
) `  ( R `  ( `' G `  B ) ) ) )
591, 2, 3, 4, 5frecuzrdgrclt 10401 . . . . . . . . . . . 12  |-  ( ph  ->  R : om --> ( (
ZZ>= `  C )  X.  S ) )
6059adantr 276 . . . . . . . . . . 11  |-  ( (
ph  /\  B  e.  ( ZZ>= `  C )
)  ->  R : om
--> ( ( ZZ>= `  C
)  X.  S ) )
6160, 52ffvelcdmd 5648 . . . . . . . . . 10  |-  ( (
ph  /\  B  e.  ( ZZ>= `  C )
)  ->  ( R `  ( `' G `  B ) )  e.  ( ( ZZ>= `  C
)  X.  S ) )
62 1st2nd2 6170 . . . . . . . . . 10  |-  ( ( R `  ( `' G `  B ) )  e.  ( (
ZZ>= `  C )  X.  S )  ->  ( R `  ( `' G `  B )
)  =  <. ( 1st `  ( R `  ( `' G `  B ) ) ) ,  ( 2nd `  ( R `
 ( `' G `  B ) ) )
>. )
6361, 62syl 14 . . . . . . . . 9  |-  ( (
ph  /\  B  e.  ( ZZ>= `  C )
)  ->  ( R `  ( `' G `  B ) )  = 
<. ( 1st `  ( R `  ( `' G `  B )
) ) ,  ( 2nd `  ( R `
 ( `' G `  B ) ) )
>. )
641adantr 276 . . . . . . . . . . . 12  |-  ( (
ph  /\  B  e.  ( ZZ>= `  C )
)  ->  C  e.  ZZ )
652adantr 276 . . . . . . . . . . . 12  |-  ( (
ph  /\  B  e.  ( ZZ>= `  C )
)  ->  A  e.  S )
663adantr 276 . . . . . . . . . . . 12  |-  ( (
ph  /\  B  e.  ( ZZ>= `  C )
)  ->  S  C_  T
)
674adantlr 477 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  B  e.  ( ZZ>= `  C )
)  /\  ( x  e.  ( ZZ>= `  C )  /\  y  e.  S
) )  ->  (
x F y )  e.  S )
6864, 65, 66, 67, 5, 52, 49frecuzrdgg 10402 . . . . . . . . . . 11  |-  ( (
ph  /\  B  e.  ( ZZ>= `  C )
)  ->  ( 1st `  ( R `  ( `' G `  B ) ) )  =  ( G `  ( `' G `  B ) ) )
69 f1ocnvfv2 5773 . . . . . . . . . . . 12  |-  ( ( G : om -1-1-onto-> ( ZZ>= `  C )  /\  B  e.  ( ZZ>=
`  C ) )  ->  ( G `  ( `' G `  B ) )  =  B )
7050, 69sylan 283 . . . . . . . . . . 11  |-  ( (
ph  /\  B  e.  ( ZZ>= `  C )
)  ->  ( G `  ( `' G `  B ) )  =  B )
7168, 70eqtrd 2210 . . . . . . . . . 10  |-  ( (
ph  /\  B  e.  ( ZZ>= `  C )
)  ->  ( 1st `  ( R `  ( `' G `  B ) ) )  =  B )
7271opeq1d 3782 . . . . . . . . 9  |-  ( (
ph  /\  B  e.  ( ZZ>= `  C )
)  ->  <. ( 1st `  ( R `  ( `' G `  B ) ) ) ,  ( 2nd `  ( R `
 ( `' G `  B ) ) )
>.  =  <. B , 
( 2nd `  ( R `  ( `' G `  B )
) ) >. )
7363, 72eqtrd 2210 . . . . . . . 8  |-  ( (
ph  /\  B  e.  ( ZZ>= `  C )
)  ->  ( R `  ( `' G `  B ) )  = 
<. B ,  ( 2nd `  ( R `  ( `' G `  B ) ) ) >. )
7473fveq2d 5515 . . . . . . 7  |-  ( (
ph  /\  B  e.  ( ZZ>= `  C )
)  ->  ( (
x  e.  ( ZZ>= `  C ) ,  y  e.  T  |->  <. (
x  +  1 ) ,  ( x F y ) >. ) `  ( R `  ( `' G `  B ) ) )  =  ( ( x  e.  (
ZZ>= `  C ) ,  y  e.  T  |->  <.
( x  +  1 ) ,  ( x F y ) >.
) `  <. B , 
( 2nd `  ( R `  ( `' G `  B )
) ) >. )
)
7558, 74eqtrd 2210 . . . . . 6  |-  ( (
ph  /\  B  e.  ( ZZ>= `  C )
)  ->  ( R `  suc  ( `' G `  B ) )  =  ( ( x  e.  ( ZZ>= `  C ) ,  y  e.  T  |-> 
<. ( x  +  1 ) ,  ( x F y ) >.
) `  <. B , 
( 2nd `  ( R `  ( `' G `  B )
) ) >. )
)
76 df-ov 5872 . . . . . 6  |-  ( B ( x  e.  (
ZZ>= `  C ) ,  y  e.  T  |->  <.
( x  +  1 ) ,  ( x F y ) >.
) ( 2nd `  ( R `  ( `' G `  B )
) ) )  =  ( ( x  e.  ( ZZ>= `  C ) ,  y  e.  T  |-> 
<. ( x  +  1 ) ,  ( x F y ) >.
) `  <. B , 
( 2nd `  ( R `  ( `' G `  B )
) ) >. )
7775, 76eqtr4di 2228 . . . . 5  |-  ( (
ph  /\  B  e.  ( ZZ>= `  C )
)  ->  ( R `  suc  ( `' G `  B ) )  =  ( B ( x  e.  ( ZZ>= `  C
) ,  y  e.  T  |->  <. ( x  + 
1 ) ,  ( x F y )
>. ) ( 2nd `  ( R `  ( `' G `  B )
) ) ) )
78 simpr 110 . . . . . 6  |-  ( (
ph  /\  B  e.  ( ZZ>= `  C )
)  ->  B  e.  ( ZZ>= `  C )
)
79 xp2nd 6161 . . . . . . . 8  |-  ( ( R `  ( `' G `  B ) )  e.  ( (
ZZ>= `  C )  X.  S )  ->  ( 2nd `  ( R `  ( `' G `  B ) ) )  e.  S
)
8061, 79syl 14 . . . . . . 7  |-  ( (
ph  /\  B  e.  ( ZZ>= `  C )
)  ->  ( 2nd `  ( R `  ( `' G `  B ) ) )  e.  S
)
8166, 80sseldd 3156 . . . . . 6  |-  ( (
ph  /\  B  e.  ( ZZ>= `  C )
)  ->  ( 2nd `  ( R `  ( `' G `  B ) ) )  e.  T
)
82 peano2uz 9572 . . . . . . . 8  |-  ( B  e.  ( ZZ>= `  C
)  ->  ( B  +  1 )  e.  ( ZZ>= `  C )
)
8382adantl 277 . . . . . . 7  |-  ( (
ph  /\  B  e.  ( ZZ>= `  C )
)  ->  ( B  +  1 )  e.  ( ZZ>= `  C )
)
8467, 78, 80caovcld 6022 . . . . . . 7  |-  ( (
ph  /\  B  e.  ( ZZ>= `  C )
)  ->  ( B F ( 2nd `  ( R `  ( `' G `  B )
) ) )  e.  S )
85 opelxp 4653 . . . . . . 7  |-  ( <.
( B  +  1 ) ,  ( B F ( 2nd `  ( R `  ( `' G `  B )
) ) ) >.  e.  ( ( ZZ>= `  C
)  X.  S )  <-> 
( ( B  + 
1 )  e.  (
ZZ>= `  C )  /\  ( B F ( 2nd `  ( R `  ( `' G `  B ) ) ) )  e.  S ) )
8683, 84, 85sylanbrc 417 . . . . . 6  |-  ( (
ph  /\  B  e.  ( ZZ>= `  C )
)  ->  <. ( B  +  1 ) ,  ( B F ( 2nd `  ( R `
 ( `' G `  B ) ) ) ) >.  e.  (
( ZZ>= `  C )  X.  S ) )
87 oveq1 5876 . . . . . . . 8  |-  ( x  =  B  ->  (
x  +  1 )  =  ( B  + 
1 ) )
88 oveq1 5876 . . . . . . . 8  |-  ( x  =  B  ->  (
x F y )  =  ( B F y ) )
8987, 88opeq12d 3784 . . . . . . 7  |-  ( x  =  B  ->  <. (
x  +  1 ) ,  ( x F y ) >.  =  <. ( B  +  1 ) ,  ( B F y ) >. )
90 oveq2 5877 . . . . . . . 8  |-  ( y  =  ( 2nd `  ( R `  ( `' G `  B )
) )  ->  ( B F y )  =  ( B F ( 2nd `  ( R `
 ( `' G `  B ) ) ) ) )
9190opeq2d 3783 . . . . . . 7  |-  ( y  =  ( 2nd `  ( R `  ( `' G `  B )
) )  ->  <. ( B  +  1 ) ,  ( B F y ) >.  =  <. ( B  +  1 ) ,  ( B F ( 2nd `  ( R `  ( `' G `  B )
) ) ) >.
)
9289, 91, 38ovmpog 6003 . . . . . 6  |-  ( ( B  e.  ( ZZ>= `  C )  /\  ( 2nd `  ( R `  ( `' G `  B ) ) )  e.  T  /\  <. ( B  + 
1 ) ,  ( B F ( 2nd `  ( R `  ( `' G `  B ) ) ) ) >.  e.  ( ( ZZ>= `  C
)  X.  S ) )  ->  ( B
( x  e.  (
ZZ>= `  C ) ,  y  e.  T  |->  <.
( x  +  1 ) ,  ( x F y ) >.
) ( 2nd `  ( R `  ( `' G `  B )
) ) )  = 
<. ( B  +  1 ) ,  ( B F ( 2nd `  ( R `  ( `' G `  B )
) ) ) >.
)
9378, 81, 86, 92syl3anc 1238 . . . . 5  |-  ( (
ph  /\  B  e.  ( ZZ>= `  C )
)  ->  ( B
( x  e.  (
ZZ>= `  C ) ,  y  e.  T  |->  <.
( x  +  1 ) ,  ( x F y ) >.
) ( 2nd `  ( R `  ( `' G `  B )
) ) )  = 
<. ( B  +  1 ) ,  ( B F ( 2nd `  ( R `  ( `' G `  B )
) ) ) >.
)
9477, 93eqtrd 2210 . . . 4  |-  ( (
ph  /\  B  e.  ( ZZ>= `  C )
)  ->  ( R `  suc  ( `' G `  B ) )  = 
<. ( B  +  1 ) ,  ( B F ( 2nd `  ( R `  ( `' G `  B )
) ) ) >.
)
95 ffun 5364 . . . . . . 7  |-  ( R : om --> ( (
ZZ>= `  C )  X.  S )  ->  Fun  R )
9660, 95syl 14 . . . . . 6  |-  ( (
ph  /\  B  e.  ( ZZ>= `  C )
)  ->  Fun  R )
97 peano2 4591 . . . . . . . 8  |-  ( ( `' G `  B )  e.  om  ->  suc  ( `' G `  B )  e.  om )
9852, 97syl 14 . . . . . . 7  |-  ( (
ph  /\  B  e.  ( ZZ>= `  C )
)  ->  suc  ( `' G `  B )  e.  om )
99 fdm 5367 . . . . . . . 8  |-  ( R : om --> ( (
ZZ>= `  C )  X.  S )  ->  dom  R  =  om )
10060, 99syl 14 . . . . . . 7  |-  ( (
ph  /\  B  e.  ( ZZ>= `  C )
)  ->  dom  R  =  om )
10198, 100eleqtrrd 2257 . . . . . 6  |-  ( (
ph  /\  B  e.  ( ZZ>= `  C )
)  ->  suc  ( `' G `  B )  e.  dom  R )
102 fvelrn 5643 . . . . . 6  |-  ( ( Fun  R  /\  suc  ( `' G `  B )  e.  dom  R )  ->  ( R `  suc  ( `' G `  B ) )  e. 
ran  R )
10396, 101, 102syl2anc 411 . . . . 5  |-  ( (
ph  /\  B  e.  ( ZZ>= `  C )
)  ->  ( R `  suc  ( `' G `  B ) )  e. 
ran  R )
1046adantr 276 . . . . 5  |-  ( (
ph  /\  B  e.  ( ZZ>= `  C )
)  ->  P  =  ran  R )
105103, 104eleqtrrd 2257 . . . 4  |-  ( (
ph  /\  B  e.  ( ZZ>= `  C )
)  ->  ( R `  suc  ( `' G `  B ) )  e.  P )
10694, 105eqeltrrd 2255 . . 3  |-  ( (
ph  /\  B  e.  ( ZZ>= `  C )
)  ->  <. ( B  +  1 ) ,  ( B F ( 2nd `  ( R `
 ( `' G `  B ) ) ) ) >.  e.  P
)
107 funopfv 5551 . . 3  |-  ( Fun 
P  ->  ( <. ( B  +  1 ) ,  ( B F ( 2nd `  ( R `  ( `' G `  B )
) ) ) >.  e.  P  ->  ( P `
 ( B  + 
1 ) )  =  ( B F ( 2nd `  ( R `
 ( `' G `  B ) ) ) ) ) )
10810, 106, 107sylc 62 . 2  |-  ( (
ph  /\  B  e.  ( ZZ>= `  C )
)  ->  ( P `  ( B  +  1 ) )  =  ( B F ( 2nd `  ( R `  ( `' G `  B ) ) ) ) )
10952, 100eleqtrrd 2257 . . . . . . 7  |-  ( (
ph  /\  B  e.  ( ZZ>= `  C )
)  ->  ( `' G `  B )  e.  dom  R )
110 fvelrn 5643 . . . . . . 7  |-  ( ( Fun  R  /\  ( `' G `  B )  e.  dom  R )  ->  ( R `  ( `' G `  B ) )  e.  ran  R
)
11196, 109, 110syl2anc 411 . . . . . 6  |-  ( (
ph  /\  B  e.  ( ZZ>= `  C )
)  ->  ( R `  ( `' G `  B ) )  e. 
ran  R )
112111, 104eleqtrrd 2257 . . . . 5  |-  ( (
ph  /\  B  e.  ( ZZ>= `  C )
)  ->  ( R `  ( `' G `  B ) )  e.  P )
11373, 112eqeltrrd 2255 . . . 4  |-  ( (
ph  /\  B  e.  ( ZZ>= `  C )
)  ->  <. B , 
( 2nd `  ( R `  ( `' G `  B )
) ) >.  e.  P
)
114 funopfv 5551 . . . 4  |-  ( Fun 
P  ->  ( <. B ,  ( 2nd `  ( R `  ( `' G `  B )
) ) >.  e.  P  ->  ( P `  B
)  =  ( 2nd `  ( R `  ( `' G `  B ) ) ) ) )
11510, 113, 114sylc 62 . . 3  |-  ( (
ph  /\  B  e.  ( ZZ>= `  C )
)  ->  ( P `  B )  =  ( 2nd `  ( R `
 ( `' G `  B ) ) ) )
116115oveq2d 5885 . 2  |-  ( (
ph  /\  B  e.  ( ZZ>= `  C )
)  ->  ( B F ( P `  B ) )  =  ( B F ( 2nd `  ( R `
 ( `' G `  B ) ) ) ) )
117108, 116eqtr4d 2213 1  |-  ( (
ph  /\  B  e.  ( ZZ>= `  C )
)  ->  ( P `  ( B  +  1 ) )  =  ( B F ( P `
 B ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    = wceq 1353    e. wcel 2148   A.wral 2455    C_ wss 3129   <.cop 3594    |-> cmpt 4061   suc csuc 4362   omcom 4586    X. cxp 4621   `'ccnv 4622   dom cdm 4623   ran crn 4624   Fun wfun 5206   -->wf 5208   -1-1-onto->wf1o 5211   ` cfv 5212  (class class class)co 5869    e. cmpo 5871   1stc1st 6133   2ndc2nd 6134  freccfrec 6385   1c1 7803    + caddc 7805   ZZcz 9242   ZZ>=cuz 9517
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4115  ax-sep 4118  ax-nul 4126  ax-pow 4171  ax-pr 4206  ax-un 4430  ax-setind 4533  ax-iinf 4584  ax-cnex 7893  ax-resscn 7894  ax-1cn 7895  ax-1re 7896  ax-icn 7897  ax-addcl 7898  ax-addrcl 7899  ax-mulcl 7900  ax-addcom 7902  ax-addass 7904  ax-distr 7906  ax-i2m1 7907  ax-0lt1 7908  ax-0id 7910  ax-rnegex 7911  ax-cnre 7913  ax-pre-ltirr 7914  ax-pre-ltwlin 7915  ax-pre-lttrn 7916  ax-pre-ltadd 7918
This theorem depends on definitions:  df-bi 117  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-int 3843  df-iun 3886  df-br 4001  df-opab 4062  df-mpt 4063  df-tr 4099  df-id 4290  df-iord 4363  df-on 4365  df-ilim 4366  df-suc 4368  df-iom 4587  df-xp 4629  df-rel 4630  df-cnv 4631  df-co 4632  df-dm 4633  df-rn 4634  df-res 4635  df-ima 4636  df-iota 5174  df-fun 5214  df-fn 5215  df-f 5216  df-f1 5217  df-fo 5218  df-f1o 5219  df-fv 5220  df-riota 5825  df-ov 5872  df-oprab 5873  df-mpo 5874  df-1st 6135  df-2nd 6136  df-recs 6300  df-frec 6386  df-pnf 7984  df-mnf 7985  df-xr 7986  df-ltxr 7987  df-le 7988  df-sub 8120  df-neg 8121  df-inn 8909  df-n0 9166  df-z 9243  df-uz 9518
This theorem is referenced by:  frecuzrdgsuct  10410
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