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Theorem fucsect 13848
Description: Two natural transformations are in a section iff all the components are in a section relation. (Contributed by Mario Carneiro, 28-Jan-2017.)
Hypotheses
Ref Expression
fuciso.q  |-  Q  =  ( C FuncCat  D )
fuciso.b  |-  B  =  ( Base `  C
)
fuciso.n  |-  N  =  ( C Nat  D )
fuciso.f  |-  ( ph  ->  F  e.  ( C 
Func  D ) )
fuciso.g  |-  ( ph  ->  G  e.  ( C 
Func  D ) )
fucsect.s  |-  S  =  (Sect `  Q )
fucsect.t  |-  T  =  (Sect `  D )
Assertion
Ref Expression
fucsect  |-  ( ph  ->  ( U ( F S G ) V  <-> 
( U  e.  ( F N G )  /\  V  e.  ( G N F )  /\  A. x  e.  B  ( U `  x ) ( ( ( 1st `  F
) `  x ) T ( ( 1st `  G ) `  x
) ) ( V `
 x ) ) ) )
Distinct variable groups:    x, B    x, C    x, D    x, F    x, G    x, N    x, V    ph, x    x, Q    x, U
Allowed substitution hints:    S( x)    T( x)

Proof of Theorem fucsect
StepHypRef Expression
1 fuciso.q . . . 4  |-  Q  =  ( C FuncCat  D )
21fucbas 13836 . . 3  |-  ( C 
Func  D )  =  (
Base `  Q )
3 fuciso.n . . . 4  |-  N  =  ( C Nat  D )
41, 3fuchom 13837 . . 3  |-  N  =  (  Hom  `  Q
)
5 eqid 2285 . . 3  |-  (comp `  Q )  =  (comp `  Q )
6 eqid 2285 . . 3  |-  ( Id
`  Q )  =  ( Id `  Q
)
7 fucsect.s . . 3  |-  S  =  (Sect `  Q )
8 fuciso.f . . . . . 6  |-  ( ph  ->  F  e.  ( C 
Func  D ) )
9 funcrcl 13739 . . . . . 6  |-  ( F  e.  ( C  Func  D )  ->  ( C  e.  Cat  /\  D  e. 
Cat ) )
108, 9syl 15 . . . . 5  |-  ( ph  ->  ( C  e.  Cat  /\  D  e.  Cat )
)
1110simpld 445 . . . 4  |-  ( ph  ->  C  e.  Cat )
1210simprd 449 . . . 4  |-  ( ph  ->  D  e.  Cat )
131, 11, 12fuccat 13846 . . 3  |-  ( ph  ->  Q  e.  Cat )
14 fuciso.g . . 3  |-  ( ph  ->  G  e.  ( C 
Func  D ) )
152, 4, 5, 6, 7, 13, 8, 14issect 13658 . 2  |-  ( ph  ->  ( U ( F S G ) V  <-> 
( U  e.  ( F N G )  /\  V  e.  ( G N F )  /\  ( V (
<. F ,  G >. (comp `  Q ) F ) U )  =  ( ( Id `  Q
) `  F )
) ) )
16 ovex 5885 . . . . . . 7  |-  ( ( V `  x ) ( <. ( ( 1st `  F ) `  x
) ,  ( ( 1st `  G ) `
 x ) >.
(comp `  D )
( ( 1st `  F
) `  x )
) ( U `  x ) )  e. 
_V
1716rgenw 2612 . . . . . 6  |-  A. x  e.  B  ( ( V `  x )
( <. ( ( 1st `  F ) `  x
) ,  ( ( 1st `  G ) `
 x ) >.
(comp `  D )
( ( 1st `  F
) `  x )
) ( U `  x ) )  e. 
_V
18 mpteqb 5616 . . . . . 6  |-  ( A. x  e.  B  (
( V `  x
) ( <. (
( 1st `  F
) `  x ) ,  ( ( 1st `  G ) `  x
) >. (comp `  D
) ( ( 1st `  F ) `  x
) ) ( U `
 x ) )  e.  _V  ->  (
( x  e.  B  |->  ( ( V `  x ) ( <.
( ( 1st `  F
) `  x ) ,  ( ( 1st `  G ) `  x
) >. (comp `  D
) ( ( 1st `  F ) `  x
) ) ( U `
 x ) ) )  =  ( x  e.  B  |->  ( ( Id `  D ) `
 ( ( 1st `  F ) `  x
) ) )  <->  A. x  e.  B  ( ( V `  x )
( <. ( ( 1st `  F ) `  x
) ,  ( ( 1st `  G ) `
 x ) >.
(comp `  D )
( ( 1st `  F
) `  x )
) ( U `  x ) )  =  ( ( Id `  D ) `  (
( 1st `  F
) `  x )
) ) )
1917, 18mp1i 11 . . . . 5  |-  ( (
ph  /\  ( U  e.  ( F N G )  /\  V  e.  ( G N F ) ) )  -> 
( ( x  e.  B  |->  ( ( V `
 x ) (
<. ( ( 1st `  F
) `  x ) ,  ( ( 1st `  G ) `  x
) >. (comp `  D
) ( ( 1st `  F ) `  x
) ) ( U `
 x ) ) )  =  ( x  e.  B  |->  ( ( Id `  D ) `
 ( ( 1st `  F ) `  x
) ) )  <->  A. x  e.  B  ( ( V `  x )
( <. ( ( 1st `  F ) `  x
) ,  ( ( 1st `  G ) `
 x ) >.
(comp `  D )
( ( 1st `  F
) `  x )
) ( U `  x ) )  =  ( ( Id `  D ) `  (
( 1st `  F
) `  x )
) ) )
20 fuciso.b . . . . . . 7  |-  B  =  ( Base `  C
)
21 eqid 2285 . . . . . . 7  |-  (comp `  D )  =  (comp `  D )
22 simprl 732 . . . . . . 7  |-  ( (
ph  /\  ( U  e.  ( F N G )  /\  V  e.  ( G N F ) ) )  ->  U  e.  ( F N G ) )
23 simprr 733 . . . . . . 7  |-  ( (
ph  /\  ( U  e.  ( F N G )  /\  V  e.  ( G N F ) ) )  ->  V  e.  ( G N F ) )
241, 3, 20, 21, 5, 22, 23fucco 13838 . . . . . 6  |-  ( (
ph  /\  ( U  e.  ( F N G )  /\  V  e.  ( G N F ) ) )  -> 
( V ( <. F ,  G >. (comp `  Q ) F ) U )  =  ( x  e.  B  |->  ( ( V `  x
) ( <. (
( 1st `  F
) `  x ) ,  ( ( 1st `  G ) `  x
) >. (comp `  D
) ( ( 1st `  F ) `  x
) ) ( U `
 x ) ) ) )
25 eqid 2285 . . . . . . . 8  |-  ( Id
`  D )  =  ( Id `  D
)
268adantr 451 . . . . . . . 8  |-  ( (
ph  /\  ( U  e.  ( F N G )  /\  V  e.  ( G N F ) ) )  ->  F  e.  ( C  Func  D ) )
271, 6, 25, 26fucid 13847 . . . . . . 7  |-  ( (
ph  /\  ( U  e.  ( F N G )  /\  V  e.  ( G N F ) ) )  -> 
( ( Id `  Q ) `  F
)  =  ( ( Id `  D )  o.  ( 1st `  F
) ) )
2812adantr 451 . . . . . . . . . 10  |-  ( (
ph  /\  ( U  e.  ( F N G )  /\  V  e.  ( G N F ) ) )  ->  D  e.  Cat )
29 eqid 2285 . . . . . . . . . . 11  |-  ( Base `  D )  =  (
Base `  D )
3029, 25cidfn 13583 . . . . . . . . . 10  |-  ( D  e.  Cat  ->  ( Id `  D )  Fn  ( Base `  D
) )
3128, 30syl 15 . . . . . . . . 9  |-  ( (
ph  /\  ( U  e.  ( F N G )  /\  V  e.  ( G N F ) ) )  -> 
( Id `  D
)  Fn  ( Base `  D ) )
32 dffn2 5392 . . . . . . . . 9  |-  ( ( Id `  D )  Fn  ( Base `  D
)  <->  ( Id `  D ) : (
Base `  D ) --> _V )
3331, 32sylib 188 . . . . . . . 8  |-  ( (
ph  /\  ( U  e.  ( F N G )  /\  V  e.  ( G N F ) ) )  -> 
( Id `  D
) : ( Base `  D ) --> _V )
34 relfunc 13738 . . . . . . . . . . 11  |-  Rel  ( C  Func  D )
35 1st2ndbr 6171 . . . . . . . . . . 11  |-  ( ( Rel  ( C  Func  D )  /\  F  e.  ( C  Func  D
) )  ->  ( 1st `  F ) ( C  Func  D )
( 2nd `  F
) )
3634, 8, 35sylancr 644 . . . . . . . . . 10  |-  ( ph  ->  ( 1st `  F
) ( C  Func  D ) ( 2nd `  F
) )
3720, 29, 36funcf1 13742 . . . . . . . . 9  |-  ( ph  ->  ( 1st `  F
) : B --> ( Base `  D ) )
3837adantr 451 . . . . . . . 8  |-  ( (
ph  /\  ( U  e.  ( F N G )  /\  V  e.  ( G N F ) ) )  -> 
( 1st `  F
) : B --> ( Base `  D ) )
39 fcompt 5696 . . . . . . . 8  |-  ( ( ( Id `  D
) : ( Base `  D ) --> _V  /\  ( 1st `  F ) : B --> ( Base `  D ) )  -> 
( ( Id `  D )  o.  ( 1st `  F ) )  =  ( x  e.  B  |->  ( ( Id
`  D ) `  ( ( 1st `  F
) `  x )
) ) )
4033, 38, 39syl2anc 642 . . . . . . 7  |-  ( (
ph  /\  ( U  e.  ( F N G )  /\  V  e.  ( G N F ) ) )  -> 
( ( Id `  D )  o.  ( 1st `  F ) )  =  ( x  e.  B  |->  ( ( Id
`  D ) `  ( ( 1st `  F
) `  x )
) ) )
4127, 40eqtrd 2317 . . . . . 6  |-  ( (
ph  /\  ( U  e.  ( F N G )  /\  V  e.  ( G N F ) ) )  -> 
( ( Id `  Q ) `  F
)  =  ( x  e.  B  |->  ( ( Id `  D ) `
 ( ( 1st `  F ) `  x
) ) ) )
4224, 41eqeq12d 2299 . . . . 5  |-  ( (
ph  /\  ( U  e.  ( F N G )  /\  V  e.  ( G N F ) ) )  -> 
( ( V (
<. F ,  G >. (comp `  Q ) F ) U )  =  ( ( Id `  Q
) `  F )  <->  ( x  e.  B  |->  ( ( V `  x
) ( <. (
( 1st `  F
) `  x ) ,  ( ( 1st `  G ) `  x
) >. (comp `  D
) ( ( 1st `  F ) `  x
) ) ( U `
 x ) ) )  =  ( x  e.  B  |->  ( ( Id `  D ) `
 ( ( 1st `  F ) `  x
) ) ) ) )
43 eqid 2285 . . . . . . . 8  |-  (  Hom  `  D )  =  (  Hom  `  D )
44 fucsect.t . . . . . . . 8  |-  T  =  (Sect `  D )
4528adantr 451 . . . . . . . 8  |-  ( ( ( ph  /\  ( U  e.  ( F N G )  /\  V  e.  ( G N F ) ) )  /\  x  e.  B )  ->  D  e.  Cat )
4638ffvelrnda 5667 . . . . . . . 8  |-  ( ( ( ph  /\  ( U  e.  ( F N G )  /\  V  e.  ( G N F ) ) )  /\  x  e.  B )  ->  ( ( 1st `  F
) `  x )  e.  ( Base `  D
) )
47 1st2ndbr 6171 . . . . . . . . . . . 12  |-  ( ( Rel  ( C  Func  D )  /\  G  e.  ( C  Func  D
) )  ->  ( 1st `  G ) ( C  Func  D )
( 2nd `  G
) )
4834, 14, 47sylancr 644 . . . . . . . . . . 11  |-  ( ph  ->  ( 1st `  G
) ( C  Func  D ) ( 2nd `  G
) )
4920, 29, 48funcf1 13742 . . . . . . . . . 10  |-  ( ph  ->  ( 1st `  G
) : B --> ( Base `  D ) )
5049adantr 451 . . . . . . . . 9  |-  ( (
ph  /\  ( U  e.  ( F N G )  /\  V  e.  ( G N F ) ) )  -> 
( 1st `  G
) : B --> ( Base `  D ) )
5150ffvelrnda 5667 . . . . . . . 8  |-  ( ( ( ph  /\  ( U  e.  ( F N G )  /\  V  e.  ( G N F ) ) )  /\  x  e.  B )  ->  ( ( 1st `  G
) `  x )  e.  ( Base `  D
) )
5222adantr 451 . . . . . . . . . 10  |-  ( ( ( ph  /\  ( U  e.  ( F N G )  /\  V  e.  ( G N F ) ) )  /\  x  e.  B )  ->  U  e.  ( F N G ) )
533, 52nat1st2nd 13827 . . . . . . . . 9  |-  ( ( ( ph  /\  ( U  e.  ( F N G )  /\  V  e.  ( G N F ) ) )  /\  x  e.  B )  ->  U  e.  ( <.
( 1st `  F
) ,  ( 2nd `  F ) >. N <. ( 1st `  G ) ,  ( 2nd `  G
) >. ) )
54 simpr 447 . . . . . . . . 9  |-  ( ( ( ph  /\  ( U  e.  ( F N G )  /\  V  e.  ( G N F ) ) )  /\  x  e.  B )  ->  x  e.  B )
553, 53, 20, 43, 54natcl 13829 . . . . . . . 8  |-  ( ( ( ph  /\  ( U  e.  ( F N G )  /\  V  e.  ( G N F ) ) )  /\  x  e.  B )  ->  ( U `  x
)  e.  ( ( ( 1st `  F
) `  x )
(  Hom  `  D ) ( ( 1st `  G
) `  x )
) )
5623adantr 451 . . . . . . . . . 10  |-  ( ( ( ph  /\  ( U  e.  ( F N G )  /\  V  e.  ( G N F ) ) )  /\  x  e.  B )  ->  V  e.  ( G N F ) )
573, 56nat1st2nd 13827 . . . . . . . . 9  |-  ( ( ( ph  /\  ( U  e.  ( F N G )  /\  V  e.  ( G N F ) ) )  /\  x  e.  B )  ->  V  e.  ( <.
( 1st `  G
) ,  ( 2nd `  G ) >. N <. ( 1st `  F ) ,  ( 2nd `  F
) >. ) )
583, 57, 20, 43, 54natcl 13829 . . . . . . . 8  |-  ( ( ( ph  /\  ( U  e.  ( F N G )  /\  V  e.  ( G N F ) ) )  /\  x  e.  B )  ->  ( V `  x
)  e.  ( ( ( 1st `  G
) `  x )
(  Hom  `  D ) ( ( 1st `  F
) `  x )
) )
5929, 43, 21, 25, 44, 45, 46, 51, 55, 58issect2 13659 . . . . . . 7  |-  ( ( ( ph  /\  ( U  e.  ( F N G )  /\  V  e.  ( G N F ) ) )  /\  x  e.  B )  ->  ( ( U `  x ) ( ( ( 1st `  F
) `  x ) T ( ( 1st `  G ) `  x
) ) ( V `
 x )  <->  ( ( V `  x )
( <. ( ( 1st `  F ) `  x
) ,  ( ( 1st `  G ) `
 x ) >.
(comp `  D )
( ( 1st `  F
) `  x )
) ( U `  x ) )  =  ( ( Id `  D ) `  (
( 1st `  F
) `  x )
) ) )
60 biidd 228 . . . . . . 7  |-  ( ( ( ph  /\  ( U  e.  ( F N G )  /\  V  e.  ( G N F ) ) )  /\  x  e.  B )  ->  ( ( ( V `
 x ) (
<. ( ( 1st `  F
) `  x ) ,  ( ( 1st `  G ) `  x
) >. (comp `  D
) ( ( 1st `  F ) `  x
) ) ( U `
 x ) )  =  ( ( Id
`  D ) `  ( ( 1st `  F
) `  x )
)  <->  ( ( V `
 x ) (
<. ( ( 1st `  F
) `  x ) ,  ( ( 1st `  G ) `  x
) >. (comp `  D
) ( ( 1st `  F ) `  x
) ) ( U `
 x ) )  =  ( ( Id
`  D ) `  ( ( 1st `  F
) `  x )
) ) )
6159, 60bitrd 244 . . . . . 6  |-  ( ( ( ph  /\  ( U  e.  ( F N G )  /\  V  e.  ( G N F ) ) )  /\  x  e.  B )  ->  ( ( U `  x ) ( ( ( 1st `  F
) `  x ) T ( ( 1st `  G ) `  x
) ) ( V `
 x )  <->  ( ( V `  x )
( <. ( ( 1st `  F ) `  x
) ,  ( ( 1st `  G ) `
 x ) >.
(comp `  D )
( ( 1st `  F
) `  x )
) ( U `  x ) )  =  ( ( Id `  D ) `  (
( 1st `  F
) `  x )
) ) )
6261ralbidva 2561 . . . . 5  |-  ( (
ph  /\  ( U  e.  ( F N G )  /\  V  e.  ( G N F ) ) )  -> 
( A. x  e.  B  ( U `  x ) ( ( ( 1st `  F
) `  x ) T ( ( 1st `  G ) `  x
) ) ( V `
 x )  <->  A. x  e.  B  ( ( V `  x )
( <. ( ( 1st `  F ) `  x
) ,  ( ( 1st `  G ) `
 x ) >.
(comp `  D )
( ( 1st `  F
) `  x )
) ( U `  x ) )  =  ( ( Id `  D ) `  (
( 1st `  F
) `  x )
) ) )
6319, 42, 623bitr4d 276 . . . 4  |-  ( (
ph  /\  ( U  e.  ( F N G )  /\  V  e.  ( G N F ) ) )  -> 
( ( V (
<. F ,  G >. (comp `  Q ) F ) U )  =  ( ( Id `  Q
) `  F )  <->  A. x  e.  B  ( U `  x ) ( ( ( 1st `  F ) `  x
) T ( ( 1st `  G ) `
 x ) ) ( V `  x
) ) )
6463pm5.32da 622 . . 3  |-  ( ph  ->  ( ( ( U  e.  ( F N G )  /\  V  e.  ( G N F ) )  /\  ( V ( <. F ,  G >. (comp `  Q
) F ) U )  =  ( ( Id `  Q ) `
 F ) )  <-> 
( ( U  e.  ( F N G )  /\  V  e.  ( G N F ) )  /\  A. x  e.  B  ( U `  x )
( ( ( 1st `  F ) `  x
) T ( ( 1st `  G ) `
 x ) ) ( V `  x
) ) ) )
65 df-3an 936 . . 3  |-  ( ( U  e.  ( F N G )  /\  V  e.  ( G N F )  /\  ( V ( <. F ,  G >. (comp `  Q
) F ) U )  =  ( ( Id `  Q ) `
 F ) )  <-> 
( ( U  e.  ( F N G )  /\  V  e.  ( G N F ) )  /\  ( V ( <. F ,  G >. (comp `  Q
) F ) U )  =  ( ( Id `  Q ) `
 F ) ) )
66 df-3an 936 . . 3  |-  ( ( U  e.  ( F N G )  /\  V  e.  ( G N F )  /\  A. x  e.  B  ( U `  x )
( ( ( 1st `  F ) `  x
) T ( ( 1st `  G ) `
 x ) ) ( V `  x
) )  <->  ( ( U  e.  ( F N G )  /\  V  e.  ( G N F ) )  /\  A. x  e.  B  ( U `  x )
( ( ( 1st `  F ) `  x
) T ( ( 1st `  G ) `
 x ) ) ( V `  x
) ) )
6764, 65, 663bitr4g 279 . 2  |-  ( ph  ->  ( ( U  e.  ( F N G )  /\  V  e.  ( G N F )  /\  ( V ( <. F ,  G >. (comp `  Q ) F ) U )  =  ( ( Id
`  Q ) `  F ) )  <->  ( U  e.  ( F N G )  /\  V  e.  ( G N F )  /\  A. x  e.  B  ( U `  x ) ( ( ( 1st `  F
) `  x ) T ( ( 1st `  G ) `  x
) ) ( V `
 x ) ) ) )
6815, 67bitrd 244 1  |-  ( ph  ->  ( U ( F S G ) V  <-> 
( U  e.  ( F N G )  /\  V  e.  ( G N F )  /\  A. x  e.  B  ( U `  x ) ( ( ( 1st `  F
) `  x ) T ( ( 1st `  G ) `  x
) ) ( V `
 x ) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1625    e. wcel 1686   A.wral 2545   _Vcvv 2790   <.cop 3645   class class class wbr 4025    e. cmpt 4079    o. ccom 4695   Rel wrel 4696    Fn wfn 5252   -->wf 5253   ` cfv 5257  (class class class)co 5860   1stc1st 6122   2ndc2nd 6123   Basecbs 13150    Hom chom 13221  compcco 13222   Catccat 13568   Idccid 13569  Sectcsect 13649    Func cfunc 13730   Nat cnat 13817   FuncCat cfuc 13818
This theorem is referenced by:  fucinv  13849
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1535  ax-5 1546  ax-17 1605  ax-9 1637  ax-8 1645  ax-13 1688  ax-14 1690  ax-6 1705  ax-7 1710  ax-11 1717  ax-12 1868  ax-ext 2266  ax-rep 4133  ax-sep 4143  ax-nul 4151  ax-pow 4190  ax-pr 4216  ax-un 4514  ax-cnex 8795  ax-resscn 8796  ax-1cn 8797  ax-icn 8798  ax-addcl 8799  ax-addrcl 8800  ax-mulcl 8801  ax-mulrcl 8802  ax-mulcom 8803  ax-addass 8804  ax-mulass 8805  ax-distr 8806  ax-i2m1 8807  ax-1ne0 8808  ax-1rid 8809  ax-rnegex 8810  ax-rrecex 8811  ax-cnre 8812  ax-pre-lttri 8813  ax-pre-lttrn 8814  ax-pre-ltadd 8815  ax-pre-mulgt0 8816
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1531  df-nf 1534  df-sb 1632  df-eu 2149  df-mo 2150  df-clab 2272  df-cleq 2278  df-clel 2281  df-nfc 2410  df-ne 2450  df-nel 2451  df-ral 2550  df-rex 2551  df-reu 2552  df-rmo 2553  df-rab 2554  df-v 2792  df-sbc 2994  df-csb 3084  df-dif 3157  df-un 3159  df-in 3161  df-ss 3168  df-pss 3170  df-nul 3458  df-if 3568  df-pw 3629  df-sn 3648  df-pr 3649  df-tp 3650  df-op 3651  df-uni 3830  df-int 3865  df-iun 3909  df-br 4026  df-opab 4080  df-mpt 4081  df-tr 4116  df-eprel 4307  df-id 4311  df-po 4316  df-so 4317  df-fr 4354  df-we 4356  df-ord 4397  df-on 4398  df-lim 4399  df-suc 4400  df-om 4659  df-xp 4697  df-rel 4698  df-cnv 4699  df-co 4700  df-dm 4701  df-rn 4702  df-res 4703  df-ima 4704  df-iota 5221  df-fun 5259  df-fn 5260  df-f 5261  df-f1 5262  df-fo 5263  df-f1o 5264  df-fv 5265  df-ov 5863  df-oprab 5864  df-mpt2 5865  df-1st 6124  df-2nd 6125  df-riota 6306  df-recs 6390  df-rdg 6425  df-1o 6481  df-oadd 6485  df-er 6662  df-map 6776  df-ixp 6820  df-en 6866  df-dom 6867  df-sdom 6868  df-fin 6869  df-pnf 8871  df-mnf 8872  df-xr 8873  df-ltxr 8874  df-le 8875  df-sub 9041  df-neg 9042  df-nn 9749  df-2 9806  df-3 9807  df-4 9808  df-5 9809  df-6 9810  df-7 9811  df-8 9812  df-9 9813  df-10 9814  df-n0 9968  df-z 10027  df-dec 10127  df-uz 10233  df-fz 10785  df-struct 13152  df-ndx 13153  df-slot 13154  df-base 13155  df-hom 13234  df-cco 13235  df-cat 13572  df-cid 13573  df-sect 13652  df-func 13734  df-nat 13819  df-fuc 13820
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