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Theorem fucsect 13842
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 13830 . . 3  |-  ( C 
Func  D )  =  (
Base `  Q )
3 fuciso.n . . . 4  |-  N  =  ( C Nat  D )
41, 3fuchom 13831 . . 3  |-  N  =  (  Hom  `  Q
)
5 eqid 2284 . . 3  |-  (comp `  Q )  =  (comp `  Q )
6 eqid 2284 . . 3  |-  ( Id
`  Q )  =  ( Id `  Q
)
7 fucsect.s . . 3  |-  S  =  (Sect `  Q )
8 fuciso.f . . . . . 6  |-  ( ph  ->  F  e.  ( C 
Func  D ) )
9 funcrcl 13733 . . . . . 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 13840 . . 3  |-  ( ph  ->  Q  e.  Cat )
14 fuciso.g . . 3  |-  ( ph  ->  G  e.  ( C 
Func  D ) )
152, 4, 5, 6, 7, 13, 8, 14issect 13652 . 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 5845 . . . . . . 7  |-  ( ( V `  x ) ( <. ( ( 1st `  F ) `  x
) ,  ( ( 1st `  G ) `
 x ) >.
(comp `  D )
( ( 1st `  F
) `  x )
) ( U `  x ) )  e. 
_V
1716rgenw 2611 . . . . . 6  |-  A. x  e.  B  ( ( V `  x )
( <. ( ( 1st `  F ) `  x
) ,  ( ( 1st `  G ) `
 x ) >.
(comp `  D )
( ( 1st `  F
) `  x )
) ( U `  x ) )  e. 
_V
18 mpteqb 5576 . . . . . 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 2284 . . . . . . 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 13832 . . . . . 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 2284 . . . . . . . 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 13841 . . . . . . 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 2284 . . . . . . . . . . 11  |-  ( Base `  D )  =  (
Base `  D )
3029, 25cidfn 13577 . . . . . . . . . 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 5356 . . . . . . . . 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 13732 . . . . . . . . . . 11  |-  Rel  ( C  Func  D )
35 1st2ndbr 6131 . . . . . . . . . . 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 13736 . . . . . . . . 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 5656 . . . . . . . 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 2316 . . . . . 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 2298 . . . . 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 2284 . . . . . . . 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 5627 . . . . . . . 8  |-  ( ( ( ph  /\  ( U  e.  ( F N G )  /\  V  e.  ( G N F ) ) )  /\  x  e.  B )  ->  ( ( 1st `  F
) `  x )  e.  ( Base `  D
) )
47 1st2ndbr 6131 . . . . . . . . . . . 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 13736 . . . . . . . . . 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 5627 . . . . . . . 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 13821 . . . . . . . . 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 13823 . . . . . . . 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 13821 . . . . . . . . 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 13823 . . . . . . . 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 13653 . . . . . . 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 2560 . . . . 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 1623    e. wcel 1685   A.wral 2544   _Vcvv 2789   <.cop 3644   class class class wbr 4024    e. cmpt 4078    o. ccom 4692   Rel wrel 4693    Fn wfn 5216   -->wf 5217   ` cfv 5221  (class class class)co 5820   1stc1st 6082   2ndc2nd 6083   Basecbs 13144    Hom chom 13215  compcco 13216   Catccat 13562   Idccid 13563  Sectcsect 13643    Func cfunc 13724   Nat cnat 13811   FuncCat cfuc 13812
This theorem is referenced by:  fucinv  13843
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1636  ax-8 1644  ax-13 1687  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1868  ax-ext 2265  ax-rep 4132  ax-sep 4142  ax-nul 4150  ax-pow 4187  ax-pr 4213  ax-un 4511  ax-cnex 8789  ax-resscn 8790  ax-1cn 8791  ax-icn 8792  ax-addcl 8793  ax-addrcl 8794  ax-mulcl 8795  ax-mulrcl 8796  ax-mulcom 8797  ax-addass 8798  ax-mulass 8799  ax-distr 8800  ax-i2m1 8801  ax-1ne0 8802  ax-1rid 8803  ax-rnegex 8804  ax-rrecex 8805  ax-cnre 8806  ax-pre-lttri 8807  ax-pre-lttrn 8808  ax-pre-ltadd 8809  ax-pre-mulgt0 8810
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 1529  df-nf 1532  df-sb 1631  df-eu 2148  df-mo 2149  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-nel 2450  df-ral 2549  df-rex 2550  df-reu 2551  df-rmo 2552  df-rab 2553  df-v 2791  df-sbc 2993  df-csb 3083  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-pss 3169  df-nul 3457  df-if 3567  df-pw 3628  df-sn 3647  df-pr 3648  df-tp 3649  df-op 3650  df-uni 3829  df-int 3864  df-iun 3908  df-br 4025  df-opab 4079  df-mpt 4080  df-tr 4115  df-eprel 4304  df-id 4308  df-po 4313  df-so 4314  df-fr 4351  df-we 4353  df-ord 4394  df-on 4395  df-lim 4396  df-suc 4397  df-om 4656  df-xp 4694  df-rel 4695  df-cnv 4696  df-co 4697  df-dm 4698  df-rn 4699  df-res 4700  df-ima 4701  df-fun 5223  df-fn 5224  df-f 5225  df-f1 5226  df-fo 5227  df-f1o 5228  df-fv 5229  df-ov 5823  df-oprab 5824  df-mpt2 5825  df-1st 6084  df-2nd 6085  df-iota 6253  df-riota 6300  df-recs 6384  df-rdg 6419  df-1o 6475  df-oadd 6479  df-er 6656  df-map 6770  df-ixp 6814  df-en 6860  df-dom 6861  df-sdom 6862  df-fin 6863  df-pnf 8865  df-mnf 8866  df-xr 8867  df-ltxr 8868  df-le 8869  df-sub 9035  df-neg 9036  df-nn 9743  df-2 9800  df-3 9801  df-4 9802  df-5 9803  df-6 9804  df-7 9805  df-8 9806  df-9 9807  df-10 9808  df-n0 9962  df-z 10021  df-dec 10121  df-uz 10227  df-fz 10779  df-struct 13146  df-ndx 13147  df-slot 13148  df-base 13149  df-hom 13228  df-cco 13229  df-cat 13566  df-cid 13567  df-sect 13646  df-func 13728  df-nat 13813  df-fuc 13814
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