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Theorem frecuzrdgsuctlem 10089
Description: Successor value of a recursive definition generator on upper integers. See comment in frec2uz0d 10065 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 10087 . . . . 5  |-  ( ph  ->  P : ( ZZ>= `  C ) --> S )
87adantr 272 . . . 4  |-  ( (
ph  /\  B  e.  ( ZZ>= `  C )
)  ->  P :
( ZZ>= `  C ) --> S )
9 ffun 5233 . . . 4  |-  ( P : ( ZZ>= `  C
) --> S  ->  Fun  P )
108, 9syl 14 . . 3  |-  ( (
ph  /\  B  e.  ( ZZ>= `  C )
)  ->  Fun  P )
11 1st2nd2 6027 . . . . . . . . . . . . . . 15  |-  ( z  e.  ( ( ZZ>= `  C )  X.  S
)  ->  z  =  <. ( 1st `  z
) ,  ( 2nd `  z ) >. )
1211adantl 273 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  B  e.  ( ZZ>= `  C )
)  /\  z  e.  ( ( ZZ>= `  C
)  X.  S ) )  ->  z  =  <. ( 1st `  z
) ,  ( 2nd `  z ) >. )
1312fveq2d 5379 . . . . . . . . . . . . 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 5731 . . . . . . . . . . . . 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, 14syl6eqr 2165 . . . . . . . . . . . 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 6017 . . . . . . . . . . . . . 14  |-  ( z  e.  ( ( ZZ>= `  C )  X.  S
)  ->  ( 1st `  z )  e.  (
ZZ>= `  C ) )
1716adantl 273 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  B  e.  ( ZZ>= `  C )
)  /\  z  e.  ( ( ZZ>= `  C
)  X.  S ) )  ->  ( 1st `  z )  e.  (
ZZ>= `  C ) )
183ad2antrr 477 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  B  e.  ( ZZ>= `  C )
)  /\  z  e.  ( ( ZZ>= `  C
)  X.  S ) )  ->  S  C_  T
)
19 xp2nd 6018 . . . . . . . . . . . . . . 15  |-  ( z  e.  ( ( ZZ>= `  C )  X.  S
)  ->  ( 2nd `  z )  e.  S
)
2019adantl 273 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  B  e.  ( ZZ>= `  C )
)  /\  z  e.  ( ( ZZ>= `  C
)  X.  S ) )  ->  ( 2nd `  z )  e.  S
)
2118, 20sseldd 3064 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  B  e.  ( ZZ>= `  C )
)  /\  z  e.  ( ( ZZ>= `  C
)  X.  S ) )  ->  ( 2nd `  z )  e.  T
)
22 peano2uz 9280 . . . . . . . . . . . . . . 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 5736 . . . . . . . . . . . . . . . 16  |-  ( y  =  ( 2nd `  z
)  ->  ( ( 1st `  z ) F y )  =  ( ( 1st `  z
) F ( 2nd `  z ) ) )
2524eleq1d 2183 . . . . . . . . . . . . . . 15  |-  ( y  =  ( 2nd `  z
)  ->  ( (
( 1st `  z
) F y )  e.  S  <->  ( ( 1st `  z ) F ( 2nd `  z
) )  e.  S
) )
26 oveq1 5735 . . . . . . . . . . . . . . . . . 18  |-  ( x  =  ( 1st `  z
)  ->  ( x F y )  =  ( ( 1st `  z
) F y ) )
2726eleq1d 2183 . . . . . . . . . . . . . . . . 17  |-  ( x  =  ( 1st `  z
)  ->  ( (
x F y )  e.  S  <->  ( ( 1st `  z ) F y )  e.  S
) )
2827ralbidv 2411 . . . . . . . . . . . . . . . 16  |-  ( x  =  ( 1st `  z
)  ->  ( A. y  e.  S  (
x F y )  e.  S  <->  A. y  e.  S  ( ( 1st `  z ) F y )  e.  S
) )
294ralrimivva 2488 . . . . . . . . . . . . . . . . 17  |-  ( ph  ->  A. x  e.  (
ZZ>= `  C ) A. y  e.  S  (
x F y )  e.  S )
3029ad2antrr 477 . . . . . . . . . . . . . . . 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 2765 . . . . . . . . . . . . . . 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 2765 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  B  e.  ( ZZ>= `  C )
)  /\  z  e.  ( ( ZZ>= `  C
)  X.  S ) )  ->  ( ( 1st `  z ) F ( 2nd `  z
) )  e.  S
)
33 opelxpi 4531 . . . . . . . . . . . . . 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 406 . . . . . . . . . . . . 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 5735 . . . . . . . . . . . . . . 15  |-  ( x  =  ( 1st `  z
)  ->  ( x  +  1 )  =  ( ( 1st `  z
)  +  1 ) )
3635, 26opeq12d 3679 . . . . . . . . . . . . . 14  |-  ( x  =  ( 1st `  z
)  ->  <. ( x  +  1 ) ,  ( x F y ) >.  =  <. ( ( 1st `  z
)  +  1 ) ,  ( ( 1st `  z ) F y ) >. )
3724opeq2d 3678 . . . . . . . . . . . . . 14  |-  ( y  =  ( 2nd `  z
)  ->  <. ( ( 1st `  z )  +  1 ) ,  ( ( 1st `  z
) F y )
>.  =  <. ( ( 1st `  z )  +  1 ) ,  ( ( 1st `  z
) F ( 2nd `  z ) ) >.
)
38 eqid 2115 . . . . . . . . . . . . . 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 5859 . . . . . . . . . . . . 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 1199 . . . . . . . . . . . 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 2147 . . . . . . . . . . 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 2191 . . . . . . . . . 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 2479 . . . . . . . . 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 9242 . . . . . . . . . . . 12  |-  ( C  e.  ZZ  ->  C  e.  ( ZZ>= `  C )
)
451, 44syl 14 . . . . . . . . . . 11  |-  ( ph  ->  C  e.  ( ZZ>= `  C ) )
46 opelxpi 4531 . . . . . . . . . . 11  |-  ( ( C  e.  ( ZZ>= `  C )  /\  A  e.  S )  ->  <. C ,  A >.  e.  ( (
ZZ>= `  C )  X.  S ) )
4745, 2, 46syl2anc 406 . . . . . . . . . 10  |-  ( ph  -> 
<. C ,  A >.  e.  ( ( ZZ>= `  C
)  X.  S ) )
4847adantr 272 . . . . . . . . 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 10072 . . . . . . . . . 10  |-  ( ph  ->  G : om -1-1-onto-> ( ZZ>= `  C )
)
51 f1ocnvdm 5636 . . . . . . . . . 10  |-  ( ( G : om -1-1-onto-> ( ZZ>= `  C )  /\  B  e.  ( ZZ>=
`  C ) )  ->  ( `' G `  B )  e.  om )
5250, 51sylan 279 . . . . . . . . 9  |-  ( (
ph  /\  B  e.  ( ZZ>= `  C )
)  ->  ( `' G `  B )  e.  om )
53 frecsuc 6258 . . . . . . . . 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 1199 . . . . . . . 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 5376 . . . . . . . 8  |-  ( R `
 suc  ( `' G `  B )
)  =  (frec ( ( x  e.  (
ZZ>= `  C ) ,  y  e.  T  |->  <.
( x  +  1 ) ,  ( x F y ) >.
) ,  <. C ,  A >. ) `  suc  ( `' G `  B ) )
565fveq1i 5376 . . . . . . . . 9  |-  ( R `
 ( `' G `  B ) )  =  (frec ( ( x  e.  ( ZZ>= `  C
) ,  y  e.  T  |->  <. ( x  + 
1 ) ,  ( x F y )
>. ) ,  <. C ,  A >. ) `  ( `' G `  B ) )
5756fveq2i 5378 . . . . . . . 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 2172 . . . . . . 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 10081 . . . . . . . . . . . 12  |-  ( ph  ->  R : om --> ( (
ZZ>= `  C )  X.  S ) )
6059adantr 272 . . . . . . . . . . 11  |-  ( (
ph  /\  B  e.  ( ZZ>= `  C )
)  ->  R : om
--> ( ( ZZ>= `  C
)  X.  S ) )
6160, 52ffvelrnd 5510 . . . . . . . . . 10  |-  ( (
ph  /\  B  e.  ( ZZ>= `  C )
)  ->  ( R `  ( `' G `  B ) )  e.  ( ( ZZ>= `  C
)  X.  S ) )
62 1st2nd2 6027 . . . . . . . . . 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 272 . . . . . . . . . . . 12  |-  ( (
ph  /\  B  e.  ( ZZ>= `  C )
)  ->  C  e.  ZZ )
652adantr 272 . . . . . . . . . . . 12  |-  ( (
ph  /\  B  e.  ( ZZ>= `  C )
)  ->  A  e.  S )
663adantr 272 . . . . . . . . . . . 12  |-  ( (
ph  /\  B  e.  ( ZZ>= `  C )
)  ->  S  C_  T
)
674adantlr 466 . . . . . . . . . . . 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 10082 . . . . . . . . . . 11  |-  ( (
ph  /\  B  e.  ( ZZ>= `  C )
)  ->  ( 1st `  ( R `  ( `' G `  B ) ) )  =  ( G `  ( `' G `  B ) ) )
69 f1ocnvfv2 5633 . . . . . . . . . . . 12  |-  ( ( G : om -1-1-onto-> ( ZZ>= `  C )  /\  B  e.  ( ZZ>=
`  C ) )  ->  ( G `  ( `' G `  B ) )  =  B )
7050, 69sylan 279 . . . . . . . . . . 11  |-  ( (
ph  /\  B  e.  ( ZZ>= `  C )
)  ->  ( G `  ( `' G `  B ) )  =  B )
7168, 70eqtrd 2147 . . . . . . . . . 10  |-  ( (
ph  /\  B  e.  ( ZZ>= `  C )
)  ->  ( 1st `  ( R `  ( `' G `  B ) ) )  =  B )
7271opeq1d 3677 . . . . . . . . 9  |-  ( (
ph  /\  B  e.  ( ZZ>= `  C )
)  ->  <. ( 1st `  ( R `  ( `' G `  B ) ) ) ,  ( 2nd `  ( R `
 ( `' G `  B ) ) )
>.  =  <. B , 
( 2nd `  ( R `  ( `' G `  B )
) ) >. )
7363, 72eqtrd 2147 . . . . . . . 8  |-  ( (
ph  /\  B  e.  ( ZZ>= `  C )
)  ->  ( R `  ( `' G `  B ) )  = 
<. B ,  ( 2nd `  ( R `  ( `' G `  B ) ) ) >. )
7473fveq2d 5379 . . . . . . 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 2147 . . . . . 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 5731 . . . . . 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, 76syl6eqr 2165 . . . . 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 109 . . . . . 6  |-  ( (
ph  /\  B  e.  ( ZZ>= `  C )
)  ->  B  e.  ( ZZ>= `  C )
)
79 xp2nd 6018 . . . . . . . 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 3064 . . . . . 6  |-  ( (
ph  /\  B  e.  ( ZZ>= `  C )
)  ->  ( 2nd `  ( R `  ( `' G `  B ) ) )  e.  T
)
82 peano2uz 9280 . . . . . . . 8  |-  ( B  e.  ( ZZ>= `  C
)  ->  ( B  +  1 )  e.  ( ZZ>= `  C )
)
8382adantl 273 . . . . . . 7  |-  ( (
ph  /\  B  e.  ( ZZ>= `  C )
)  ->  ( B  +  1 )  e.  ( ZZ>= `  C )
)
8467, 78, 80caovcld 5878 . . . . . . 7  |-  ( (
ph  /\  B  e.  ( ZZ>= `  C )
)  ->  ( B F ( 2nd `  ( R `  ( `' G `  B )
) ) )  e.  S )
85 opelxp 4529 . . . . . . 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 411 . . . . . 6  |-  ( (
ph  /\  B  e.  ( ZZ>= `  C )
)  ->  <. ( B  +  1 ) ,  ( B F ( 2nd `  ( R `
 ( `' G `  B ) ) ) ) >.  e.  (
( ZZ>= `  C )  X.  S ) )
87 oveq1 5735 . . . . . . . 8  |-  ( x  =  B  ->  (
x  +  1 )  =  ( B  + 
1 ) )
88 oveq1 5735 . . . . . . . 8  |-  ( x  =  B  ->  (
x F y )  =  ( B F y ) )
8987, 88opeq12d 3679 . . . . . . 7  |-  ( x  =  B  ->  <. (
x  +  1 ) ,  ( x F y ) >.  =  <. ( B  +  1 ) ,  ( B F y ) >. )
90 oveq2 5736 . . . . . . . 8  |-  ( y  =  ( 2nd `  ( R `  ( `' G `  B )
) )  ->  ( B F y )  =  ( B F ( 2nd `  ( R `
 ( `' G `  B ) ) ) ) )
9190opeq2d 3678 . . . . . . 7  |-  ( y  =  ( 2nd `  ( R `  ( `' G `  B )
) )  ->  <. ( B  +  1 ) ,  ( B F y ) >.  =  <. ( B  +  1 ) ,  ( B F ( 2nd `  ( R `  ( `' G `  B )
) ) ) >.
)
9289, 91, 38ovmpog 5859 . . . . . 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 1199 . . . . 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 2147 . . . 4  |-  ( (
ph  /\  B  e.  ( ZZ>= `  C )
)  ->  ( R `  suc  ( `' G `  B ) )  = 
<. ( B  +  1 ) ,  ( B F ( 2nd `  ( R `  ( `' G `  B )
) ) ) >.
)
95 ffun 5233 . . . . . . 7  |-  ( R : om --> ( (
ZZ>= `  C )  X.  S )  ->  Fun  R )
9660, 95syl 14 . . . . . 6  |-  ( (
ph  /\  B  e.  ( ZZ>= `  C )
)  ->  Fun  R )
97 peano2 4469 . . . . . . . 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 5236 . . . . . . . 8  |-  ( R : om --> ( (
ZZ>= `  C )  X.  S )  ->  dom  R  =  om )
10060, 99syl 14 . . . . . . 7  |-  ( (
ph  /\  B  e.  ( ZZ>= `  C )
)  ->  dom  R  =  om )
10198, 100eleqtrrd 2194 . . . . . 6  |-  ( (
ph  /\  B  e.  ( ZZ>= `  C )
)  ->  suc  ( `' G `  B )  e.  dom  R )
102 fvelrn 5505 . . . . . 6  |-  ( ( Fun  R  /\  suc  ( `' G `  B )  e.  dom  R )  ->  ( R `  suc  ( `' G `  B ) )  e. 
ran  R )
10396, 101, 102syl2anc 406 . . . . 5  |-  ( (
ph  /\  B  e.  ( ZZ>= `  C )
)  ->  ( R `  suc  ( `' G `  B ) )  e. 
ran  R )
1046adantr 272 . . . . 5  |-  ( (
ph  /\  B  e.  ( ZZ>= `  C )
)  ->  P  =  ran  R )
105103, 104eleqtrrd 2194 . . . 4  |-  ( (
ph  /\  B  e.  ( ZZ>= `  C )
)  ->  ( R `  suc  ( `' G `  B ) )  e.  P )
10694, 105eqeltrrd 2192 . . 3  |-  ( (
ph  /\  B  e.  ( ZZ>= `  C )
)  ->  <. ( B  +  1 ) ,  ( B F ( 2nd `  ( R `
 ( `' G `  B ) ) ) ) >.  e.  P
)
107 funopfv 5415 . . 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 2194 . . . . . . 7  |-  ( (
ph  /\  B  e.  ( ZZ>= `  C )
)  ->  ( `' G `  B )  e.  dom  R )
110 fvelrn 5505 . . . . . . 7  |-  ( ( Fun  R  /\  ( `' G `  B )  e.  dom  R )  ->  ( R `  ( `' G `  B ) )  e.  ran  R
)
11196, 109, 110syl2anc 406 . . . . . 6  |-  ( (
ph  /\  B  e.  ( ZZ>= `  C )
)  ->  ( R `  ( `' G `  B ) )  e. 
ran  R )
112111, 104eleqtrrd 2194 . . . . 5  |-  ( (
ph  /\  B  e.  ( ZZ>= `  C )
)  ->  ( R `  ( `' G `  B ) )  e.  P )
11373, 112eqeltrrd 2192 . . . 4  |-  ( (
ph  /\  B  e.  ( ZZ>= `  C )
)  ->  <. B , 
( 2nd `  ( R `  ( `' G `  B )
) ) >.  e.  P
)
114 funopfv 5415 . . . 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 5744 . 2  |-  ( (
ph  /\  B  e.  ( ZZ>= `  C )
)  ->  ( B F ( P `  B ) )  =  ( B F ( 2nd `  ( R `
 ( `' G `  B ) ) ) ) )
117108, 116eqtr4d 2150 1  |-  ( (
ph  /\  B  e.  ( ZZ>= `  C )
)  ->  ( P `  ( B  +  1 ) )  =  ( B F ( P `
 B ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    = wceq 1314    e. wcel 1463   A.wral 2390    C_ wss 3037   <.cop 3496    |-> cmpt 3949   suc csuc 4247   omcom 4464    X. cxp 4497   `'ccnv 4498   dom cdm 4499   ran crn 4500   Fun wfun 5075   -->wf 5077   -1-1-onto->wf1o 5080   ` 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:  frecuzrdgsuct  10090
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