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Theorem frecuzrdgtcl 10192
Description: The recursive definition generator on upper integers is a function. See comment in frec2uz0d 10179 for the description of  G as the mapping from  om to  ( ZZ>= `  C
). (Contributed by Jim Kingdon, 26-May-2020.)
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 >. )
frecuzrdgtcl.3  |-  ( ph  ->  T  =  ran  R
)
Assertion
Ref Expression
frecuzrdgtcl  |-  ( ph  ->  T : ( ZZ>= `  C ) --> 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)    T( x, y)    G( x)

Proof of Theorem frecuzrdgtcl
Dummy variables  w  z  v are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 frecuzrdgtcl.3 . . . . . . . . . 10  |-  ( ph  ->  T  =  ran  R
)
21eleq2d 2209 . . . . . . . . 9  |-  ( ph  ->  ( z  e.  T  <->  z  e.  ran  R ) )
3 frec2uz.1 . . . . . . . . . . 11  |-  ( ph  ->  C  e.  ZZ )
4 frec2uz.2 . . . . . . . . . . 11  |-  G  = frec ( ( x  e.  ZZ  |->  ( x  + 
1 ) ) ,  C )
5 frecuzrdgrrn.a . . . . . . . . . . 11  |-  ( ph  ->  A  e.  S )
6 frecuzrdgrrn.f . . . . . . . . . . 11  |-  ( (
ph  /\  ( x  e.  ( ZZ>= `  C )  /\  y  e.  S
) )  ->  (
x F y )  e.  S )
7 frecuzrdgrrn.2 . . . . . . . . . . 11  |-  R  = frec ( ( x  e.  ( ZZ>= `  C ) ,  y  e.  S  |-> 
<. ( x  +  1 ) ,  ( x F y ) >.
) ,  <. C ,  A >. )
83, 4, 5, 6, 7frecuzrdgrcl 10190 . . . . . . . . . 10  |-  ( ph  ->  R : om --> ( (
ZZ>= `  C )  X.  S ) )
9 ffn 5272 . . . . . . . . . 10  |-  ( R : om --> ( (
ZZ>= `  C )  X.  S )  ->  R  Fn  om )
10 fvelrnb 5469 . . . . . . . . . 10  |-  ( R  Fn  om  ->  (
z  e.  ran  R  <->  E. w  e.  om  ( R `  w )  =  z ) )
118, 9, 103syl 17 . . . . . . . . 9  |-  ( ph  ->  ( z  e.  ran  R  <->  E. w  e.  om  ( R `  w )  =  z ) )
122, 11bitrd 187 . . . . . . . 8  |-  ( ph  ->  ( z  e.  T  <->  E. w  e.  om  ( R `  w )  =  z ) )
133, 4, 5, 6, 7frecuzrdgrrn 10188 . . . . . . . . . 10  |-  ( (
ph  /\  w  e.  om )  ->  ( R `  w )  e.  ( ( ZZ>= `  C )  X.  S ) )
14 eleq1 2202 . . . . . . . . . 10  |-  ( ( R `  w )  =  z  ->  (
( R `  w
)  e.  ( (
ZZ>= `  C )  X.  S )  <->  z  e.  ( ( ZZ>= `  C
)  X.  S ) ) )
1513, 14syl5ibcom 154 . . . . . . . . 9  |-  ( (
ph  /\  w  e.  om )  ->  ( ( R `  w )  =  z  ->  z  e.  ( ( ZZ>= `  C
)  X.  S ) ) )
1615rexlimdva 2549 . . . . . . . 8  |-  ( ph  ->  ( E. w  e. 
om  ( R `  w )  =  z  ->  z  e.  ( ( ZZ>= `  C )  X.  S ) ) )
1712, 16sylbid 149 . . . . . . 7  |-  ( ph  ->  ( z  e.  T  ->  z  e.  ( (
ZZ>= `  C )  X.  S ) ) )
1817ssrdv 3103 . . . . . 6  |-  ( ph  ->  T  C_  ( ( ZZ>=
`  C )  X.  S ) )
19 xpss 4647 . . . . . 6  |-  ( (
ZZ>= `  C )  X.  S )  C_  ( _V  X.  _V )
2018, 19sstrdi 3109 . . . . 5  |-  ( ph  ->  T  C_  ( _V  X.  _V ) )
21 df-rel 4546 . . . . 5  |-  ( Rel 
T  <->  T  C_  ( _V 
X.  _V ) )
2220, 21sylibr 133 . . . 4  |-  ( ph  ->  Rel  T )
233, 4frec2uzf1od 10186 . . . . . . . . . . 11  |-  ( ph  ->  G : om -1-1-onto-> ( ZZ>= `  C )
)
24 f1ocnvdm 5682 . . . . . . . . . . 11  |-  ( ( G : om -1-1-onto-> ( ZZ>= `  C )  /\  v  e.  ( ZZ>=
`  C ) )  ->  ( `' G `  v )  e.  om )
2523, 24sylan 281 . . . . . . . . . 10  |-  ( (
ph  /\  v  e.  ( ZZ>= `  C )
)  ->  ( `' G `  v )  e.  om )
263, 4, 5, 6, 7frecuzrdgrrn 10188 . . . . . . . . . 10  |-  ( (
ph  /\  ( `' G `  v )  e.  om )  ->  ( R `  ( `' G `  v )
)  e.  ( (
ZZ>= `  C )  X.  S ) )
2725, 26syldan 280 . . . . . . . . 9  |-  ( (
ph  /\  v  e.  ( ZZ>= `  C )
)  ->  ( R `  ( `' G `  v ) )  e.  ( ( ZZ>= `  C
)  X.  S ) )
28 xp2nd 6064 . . . . . . . . 9  |-  ( ( R `  ( `' G `  v ) )  e.  ( (
ZZ>= `  C )  X.  S )  ->  ( 2nd `  ( R `  ( `' G `  v ) ) )  e.  S
)
2927, 28syl 14 . . . . . . . 8  |-  ( (
ph  /\  v  e.  ( ZZ>= `  C )
)  ->  ( 2nd `  ( R `  ( `' G `  v ) ) )  e.  S
)
301eleq2d 2209 . . . . . . . . . . . 12  |-  ( ph  ->  ( <. v ,  z
>.  e.  T  <->  <. v ,  z >.  e.  ran  R ) )
31 fvelrnb 5469 . . . . . . . . . . . . 13  |-  ( R  Fn  om  ->  ( <. v ,  z >.  e.  ran  R  <->  E. w  e.  om  ( R `  w )  =  <. v ,  z >. )
)
328, 9, 313syl 17 . . . . . . . . . . . 12  |-  ( ph  ->  ( <. v ,  z
>.  e.  ran  R  <->  E. w  e.  om  ( R `  w )  =  <. v ,  z >. )
)
3330, 32bitrd 187 . . . . . . . . . . 11  |-  ( ph  ->  ( <. v ,  z
>.  e.  T  <->  E. w  e.  om  ( R `  w )  =  <. v ,  z >. )
)
343adantr 274 . . . . . . . . . . . . . . . . . . . 20  |-  ( (
ph  /\  w  e.  om )  ->  C  e.  ZZ )
355adantr 274 . . . . . . . . . . . . . . . . . . . 20  |-  ( (
ph  /\  w  e.  om )  ->  A  e.  S )
366adantlr 468 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( ph  /\  w  e.  om )  /\  (
x  e.  ( ZZ>= `  C )  /\  y  e.  S ) )  -> 
( x F y )  e.  S )
37 simpr 109 . . . . . . . . . . . . . . . . . . . 20  |-  ( (
ph  /\  w  e.  om )  ->  w  e.  om )
3834, 4, 35, 36, 7, 37frec2uzrdg 10189 . . . . . . . . . . . . . . . . . . 19  |-  ( (
ph  /\  w  e.  om )  ->  ( R `  w )  =  <. ( G `  w ) ,  ( 2nd `  ( R `  w )
) >. )
3938eqeq1d 2148 . . . . . . . . . . . . . . . . . 18  |-  ( (
ph  /\  w  e.  om )  ->  ( ( R `  w )  =  <. v ,  z
>. 
<-> 
<. ( G `  w
) ,  ( 2nd `  ( R `  w
) ) >.  =  <. v ,  z >. )
)
40 vex 2689 . . . . . . . . . . . . . . . . . . . 20  |-  v  e. 
_V
41 vex 2689 . . . . . . . . . . . . . . . . . . . 20  |-  z  e. 
_V
4240, 41opth2 4162 . . . . . . . . . . . . . . . . . . 19  |-  ( <.
( G `  w
) ,  ( 2nd `  ( R `  w
) ) >.  =  <. v ,  z >.  <->  ( ( G `  w )  =  v  /\  ( 2nd `  ( R `  w ) )  =  z ) )
4342simplbi 272 . . . . . . . . . . . . . . . . . 18  |-  ( <.
( G `  w
) ,  ( 2nd `  ( R `  w
) ) >.  =  <. v ,  z >.  ->  ( G `  w )  =  v )
4439, 43syl6bi 162 . . . . . . . . . . . . . . . . 17  |-  ( (
ph  /\  w  e.  om )  ->  ( ( R `  w )  =  <. v ,  z
>.  ->  ( G `  w )  =  v ) )
45 f1ocnvfv 5680 . . . . . . . . . . . . . . . . . 18  |-  ( ( G : om -1-1-onto-> ( ZZ>= `  C )  /\  w  e.  om )  ->  ( ( G `
 w )  =  v  ->  ( `' G `  v )  =  w ) )
4623, 45sylan 281 . . . . . . . . . . . . . . . . 17  |-  ( (
ph  /\  w  e.  om )  ->  ( ( G `  w )  =  v  ->  ( `' G `  v )  =  w ) )
4744, 46syld 45 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  w  e.  om )  ->  ( ( R `  w )  =  <. v ,  z
>.  ->  ( `' G `  v )  =  w ) )
48 fveq2 5421 . . . . . . . . . . . . . . . . 17  |-  ( ( `' G `  v )  =  w  ->  ( R `  ( `' G `  v )
)  =  ( R `
 w ) )
4948fveq2d 5425 . . . . . . . . . . . . . . . 16  |-  ( ( `' G `  v )  =  w  ->  ( 2nd `  ( R `  ( `' G `  v ) ) )  =  ( 2nd `  ( R `
 w ) ) )
5047, 49syl6 33 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  w  e.  om )  ->  ( ( R `  w )  =  <. v ,  z
>.  ->  ( 2nd `  ( R `  ( `' G `  v )
) )  =  ( 2nd `  ( R `
 w ) ) ) )
5150imp 123 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  w  e.  om )  /\  ( R `  w )  =  <. v ,  z
>. )  ->  ( 2nd `  ( R `  ( `' G `  v ) ) )  =  ( 2nd `  ( R `
 w ) ) )
5240, 41op2ndd 6047 . . . . . . . . . . . . . . 15  |-  ( ( R `  w )  =  <. v ,  z
>.  ->  ( 2nd `  ( R `  w )
)  =  z )
5352adantl 275 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  w  e.  om )  /\  ( R `  w )  =  <. v ,  z
>. )  ->  ( 2nd `  ( R `  w
) )  =  z )
5451, 53eqtr2d 2173 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  w  e.  om )  /\  ( R `  w )  =  <. v ,  z
>. )  ->  z  =  ( 2nd `  ( R `  ( `' G `  v )
) ) )
5554ex 114 . . . . . . . . . . . 12  |-  ( (
ph  /\  w  e.  om )  ->  ( ( R `  w )  =  <. v ,  z
>.  ->  z  =  ( 2nd `  ( R `
 ( `' G `  v ) ) ) ) )
5655rexlimdva 2549 . . . . . . . . . . 11  |-  ( ph  ->  ( E. w  e. 
om  ( R `  w )  =  <. v ,  z >.  ->  z  =  ( 2nd `  ( R `  ( `' G `  v )
) ) ) )
5733, 56sylbid 149 . . . . . . . . . 10  |-  ( ph  ->  ( <. v ,  z
>.  e.  T  ->  z  =  ( 2nd `  ( R `  ( `' G `  v )
) ) ) )
5857alrimiv 1846 . . . . . . . . 9  |-  ( ph  ->  A. z ( <.
v ,  z >.  e.  T  ->  z  =  ( 2nd `  ( R `  ( `' G `  v )
) ) ) )
5958adantr 274 . . . . . . . 8  |-  ( (
ph  /\  v  e.  ( ZZ>= `  C )
)  ->  A. z
( <. v ,  z
>.  e.  T  ->  z  =  ( 2nd `  ( R `  ( `' G `  v )
) ) ) )
60 eqeq2 2149 . . . . . . . . . . 11  |-  ( w  =  ( 2nd `  ( R `  ( `' G `  v )
) )  ->  (
z  =  w  <->  z  =  ( 2nd `  ( R `
 ( `' G `  v ) ) ) ) )
6160imbi2d 229 . . . . . . . . . 10  |-  ( w  =  ( 2nd `  ( R `  ( `' G `  v )
) )  ->  (
( <. v ,  z
>.  e.  T  ->  z  =  w )  <->  ( <. v ,  z >.  e.  T  ->  z  =  ( 2nd `  ( R `  ( `' G `  v ) ) ) ) ) )
6261albidv 1796 . . . . . . . . 9  |-  ( w  =  ( 2nd `  ( R `  ( `' G `  v )
) )  ->  ( A. z ( <. v ,  z >.  e.  T  ->  z  =  w )  <->  A. z ( <. v ,  z >.  e.  T  ->  z  =  ( 2nd `  ( R `  ( `' G `  v ) ) ) ) ) )
6362spcegv 2774 . . . . . . . 8  |-  ( ( 2nd `  ( R `
 ( `' G `  v ) ) )  e.  S  ->  ( A. z ( <. v ,  z >.  e.  T  ->  z  =  ( 2nd `  ( R `  ( `' G `  v ) ) ) )  ->  E. w A. z (
<. v ,  z >.  e.  T  ->  z  =  w ) ) )
6429, 59, 63sylc 62 . . . . . . 7  |-  ( (
ph  /\  v  e.  ( ZZ>= `  C )
)  ->  E. w A. z ( <. v ,  z >.  e.  T  ->  z  =  w ) )
65 nfv 1508 . . . . . . . 8  |-  F/ w <. v ,  z >.  e.  T
6665mo2r 2051 . . . . . . 7  |-  ( E. w A. z (
<. v ,  z >.  e.  T  ->  z  =  w )  ->  E* z <. v ,  z
>.  e.  T )
6764, 66syl 14 . . . . . 6  |-  ( (
ph  /\  v  e.  ( ZZ>= `  C )
)  ->  E* z <. v ,  z >.  e.  T )
68 dmss 4738 . . . . . . . . . . 11  |-  ( T 
C_  ( ( ZZ>= `  C )  X.  S
)  ->  dom  T  C_  dom  ( ( ZZ>= `  C
)  X.  S ) )
6918, 68syl 14 . . . . . . . . . 10  |-  ( ph  ->  dom  T  C_  dom  ( ( ZZ>= `  C
)  X.  S ) )
70 dmxpss 4969 . . . . . . . . . 10  |-  dom  (
( ZZ>= `  C )  X.  S )  C_  ( ZZ>=
`  C )
7169, 70sstrdi 3109 . . . . . . . . 9  |-  ( ph  ->  dom  T  C_  ( ZZ>=
`  C ) )
723adantr 274 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  v  e.  ( ZZ>= `  C )
)  ->  C  e.  ZZ )
735adantr 274 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  v  e.  ( ZZ>= `  C )
)  ->  A  e.  S )
746adantlr 468 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  v  e.  ( ZZ>= `  C )
)  /\  ( x  e.  ( ZZ>= `  C )  /\  y  e.  S
) )  ->  (
x F y )  e.  S )
75 simpr 109 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  v  e.  ( ZZ>= `  C )
)  ->  v  e.  ( ZZ>= `  C )
)
7672, 4, 73, 74, 7, 75frecuzrdglem 10191 . . . . . . . . . . . . 13  |-  ( (
ph  /\  v  e.  ( ZZ>= `  C )
)  ->  <. v ,  ( 2nd `  ( R `  ( `' G `  v )
) ) >.  e.  ran  R )
771eleq2d 2209 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( <. v ,  ( 2nd `  ( R `
 ( `' G `  v ) ) )
>.  e.  T  <->  <. v ,  ( 2nd `  ( R `  ( `' G `  v )
) ) >.  e.  ran  R ) )
7877adantr 274 . . . . . . . . . . . . 13  |-  ( (
ph  /\  v  e.  ( ZZ>= `  C )
)  ->  ( <. v ,  ( 2nd `  ( R `  ( `' G `  v )
) ) >.  e.  T  <->  <.
v ,  ( 2nd `  ( R `  ( `' G `  v ) ) ) >.  e.  ran  R ) )
7976, 78mpbird 166 . . . . . . . . . . . 12  |-  ( (
ph  /\  v  e.  ( ZZ>= `  C )
)  ->  <. v ,  ( 2nd `  ( R `  ( `' G `  v )
) ) >.  e.  T
)
80 opeldmg 4744 . . . . . . . . . . . . 13  |-  ( ( v  e.  _V  /\  ( 2nd `  ( R `
 ( `' G `  v ) ) )  e.  S )  -> 
( <. v ,  ( 2nd `  ( R `
 ( `' G `  v ) ) )
>.  e.  T  ->  v  e.  dom  T ) )
8140, 80mpan 420 . . . . . . . . . . . 12  |-  ( ( 2nd `  ( R `
 ( `' G `  v ) ) )  e.  S  ->  ( <. v ,  ( 2nd `  ( R `  ( `' G `  v ) ) ) >.  e.  T  ->  v  e.  dom  T
) )
8229, 79, 81sylc 62 . . . . . . . . . . 11  |-  ( (
ph  /\  v  e.  ( ZZ>= `  C )
)  ->  v  e.  dom  T )
8382ex 114 . . . . . . . . . 10  |-  ( ph  ->  ( v  e.  (
ZZ>= `  C )  -> 
v  e.  dom  T
) )
8483ssrdv 3103 . . . . . . . . 9  |-  ( ph  ->  ( ZZ>= `  C )  C_ 
dom  T )
8571, 84eqssd 3114 . . . . . . . 8  |-  ( ph  ->  dom  T  =  (
ZZ>= `  C ) )
8685eleq2d 2209 . . . . . . 7  |-  ( ph  ->  ( v  e.  dom  T  <-> 
v  e.  ( ZZ>= `  C ) ) )
8786pm5.32i 449 . . . . . 6  |-  ( (
ph  /\  v  e.  dom  T )  <->  ( ph  /\  v  e.  ( ZZ>= `  C ) ) )
88 df-br 3930 . . . . . . 7  |-  ( v T z  <->  <. v ,  z >.  e.  T
)
8988mobii 2036 . . . . . 6  |-  ( E* z  v T z  <->  E* z <. v ,  z
>.  e.  T )
9067, 87, 893imtr4i 200 . . . . 5  |-  ( (
ph  /\  v  e.  dom  T )  ->  E* z  v T z )
9190ralrimiva 2505 . . . 4  |-  ( ph  ->  A. v  e.  dom  T E* z  v T z )
92 dffun7 5150 . . . 4  |-  ( Fun 
T  <->  ( Rel  T  /\  A. v  e.  dom  T E* z  v T z ) )
9322, 91, 92sylanbrc 413 . . 3  |-  ( ph  ->  Fun  T )
94 df-fn 5126 . . 3  |-  ( T  Fn  ( ZZ>= `  C
)  <->  ( Fun  T  /\  dom  T  =  (
ZZ>= `  C ) ) )
9593, 85, 94sylanbrc 413 . 2  |-  ( ph  ->  T  Fn  ( ZZ>= `  C ) )
96 rnss 4769 . . . 4  |-  ( T 
C_  ( ( ZZ>= `  C )  X.  S
)  ->  ran  T  C_  ran  ( ( ZZ>= `  C
)  X.  S ) )
9718, 96syl 14 . . 3  |-  ( ph  ->  ran  T  C_  ran  ( ( ZZ>= `  C
)  X.  S ) )
98 rnxpss 4970 . . 3  |-  ran  (
( ZZ>= `  C )  X.  S )  C_  S
9997, 98sstrdi 3109 . 2  |-  ( ph  ->  ran  T  C_  S
)
100 df-f 5127 . 2  |-  ( T : ( ZZ>= `  C
) --> S  <->  ( T  Fn  ( ZZ>= `  C )  /\  ran  T  C_  S
) )
10195, 99, 100sylanbrc 413 1  |-  ( ph  ->  T : ( ZZ>= `  C ) --> S )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104   A.wal 1329    = wceq 1331   E.wex 1468    e. wcel 1480   E*wmo 2000   A.wral 2416   E.wrex 2417   _Vcvv 2686    C_ wss 3071   <.cop 3530   class class class wbr 3929    |-> cmpt 3989   omcom 4504    X. cxp 4537   `'ccnv 4538   dom cdm 4539   ran crn 4540   Rel wrel 4544   Fun wfun 5117    Fn wfn 5118   -->wf 5119   -1-1-onto->wf1o 5122   ` cfv 5123  (class class class)co 5774    e. cmpo 5776   2ndc2nd 6037  freccfrec 6287   1c1 7628    + caddc 7630   ZZcz 9061   ZZ>=cuz 9333
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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-coll 4043  ax-sep 4046  ax-nul 4054  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452  ax-iinf 4502  ax-cnex 7718  ax-resscn 7719  ax-1cn 7720  ax-1re 7721  ax-icn 7722  ax-addcl 7723  ax-addrcl 7724  ax-mulcl 7725  ax-addcom 7727  ax-addass 7729  ax-distr 7731  ax-i2m1 7732  ax-0lt1 7733  ax-0id 7735  ax-rnegex 7736  ax-cnre 7738  ax-pre-ltirr 7739  ax-pre-ltwlin 7740  ax-pre-lttrn 7741  ax-pre-ltadd 7743
This theorem depends on definitions:  df-bi 116  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-nel 2404  df-ral 2421  df-rex 2422  df-reu 2423  df-rab 2425  df-v 2688  df-sbc 2910  df-csb 3004  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-nul 3364  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-int 3772  df-iun 3815  df-br 3930  df-opab 3990  df-mpt 3991  df-tr 4027  df-id 4215  df-iord 4288  df-on 4290  df-ilim 4291  df-suc 4293  df-iom 4505  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-f1 5128  df-fo 5129  df-f1o 5130  df-fv 5131  df-riota 5730  df-ov 5777  df-oprab 5778  df-mpo 5779  df-1st 6038  df-2nd 6039  df-recs 6202  df-frec 6288  df-pnf 7809  df-mnf 7810  df-xr 7811  df-ltxr 7812  df-le 7813  df-sub 7942  df-neg 7943  df-inn 8728  df-n0 8985  df-z 9062  df-uz 9334
This theorem is referenced by:  frecuzrdg0  10193  frecuzrdgsuc  10194
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