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Theorem fucsect 14089
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 14077 . . 3  |-  ( C 
Func  D )  =  (
Base `  Q )
3 fuciso.n . . . 4  |-  N  =  ( C Nat  D )
41, 3fuchom 14078 . . 3  |-  N  =  (  Hom  `  Q
)
5 eqid 2380 . . 3  |-  (comp `  Q )  =  (comp `  Q )
6 eqid 2380 . . 3  |-  ( Id
`  Q )  =  ( Id `  Q
)
7 fucsect.s . . 3  |-  S  =  (Sect `  Q )
8 fuciso.f . . . . . 6  |-  ( ph  ->  F  e.  ( C 
Func  D ) )
9 funcrcl 13980 . . . . . 6  |-  ( F  e.  ( C  Func  D )  ->  ( C  e.  Cat  /\  D  e. 
Cat ) )
108, 9syl 16 . . . . 5  |-  ( ph  ->  ( C  e.  Cat  /\  D  e.  Cat )
)
1110simpld 446 . . . 4  |-  ( ph  ->  C  e.  Cat )
1210simprd 450 . . . 4  |-  ( ph  ->  D  e.  Cat )
131, 11, 12fuccat 14087 . . 3  |-  ( ph  ->  Q  e.  Cat )
14 fuciso.g . . 3  |-  ( ph  ->  G  e.  ( C 
Func  D ) )
152, 4, 5, 6, 7, 13, 8, 14issect 13899 . 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 6038 . . . . . . 7  |-  ( ( V `  x ) ( <. ( ( 1st `  F ) `  x
) ,  ( ( 1st `  G ) `
 x ) >.
(comp `  D )
( ( 1st `  F
) `  x )
) ( U `  x ) )  e. 
_V
1716rgenw 2709 . . . . . 6  |-  A. x  e.  B  ( ( V `  x )
( <. ( ( 1st `  F ) `  x
) ,  ( ( 1st `  G ) `
 x ) >.
(comp `  D )
( ( 1st `  F
) `  x )
) ( U `  x ) )  e. 
_V
18 mpteqb 5751 . . . . . 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 12 . . . . 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 2380 . . . . . . 7  |-  (comp `  D )  =  (comp `  D )
22 simprl 733 . . . . . . 7  |-  ( (
ph  /\  ( U  e.  ( F N G )  /\  V  e.  ( G N F ) ) )  ->  U  e.  ( F N G ) )
23 simprr 734 . . . . . . 7  |-  ( (
ph  /\  ( U  e.  ( F N G )  /\  V  e.  ( G N F ) ) )  ->  V  e.  ( G N F ) )
241, 3, 20, 21, 5, 22, 23fucco 14079 . . . . . 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 2380 . . . . . . . 8  |-  ( Id
`  D )  =  ( Id `  D
)
268adantr 452 . . . . . . . 8  |-  ( (
ph  /\  ( U  e.  ( F N G )  /\  V  e.  ( G N F ) ) )  ->  F  e.  ( C  Func  D ) )
271, 6, 25, 26fucid 14088 . . . . . . 7  |-  ( (
ph  /\  ( U  e.  ( F N G )  /\  V  e.  ( G N F ) ) )  -> 
( ( Id `  Q ) `  F
)  =  ( ( Id `  D )  o.  ( 1st `  F
) ) )
2812adantr 452 . . . . . . . . . 10  |-  ( (
ph  /\  ( U  e.  ( F N G )  /\  V  e.  ( G N F ) ) )  ->  D  e.  Cat )
29 eqid 2380 . . . . . . . . . . 11  |-  ( Base `  D )  =  (
Base `  D )
3029, 25cidfn 13824 . . . . . . . . . 10  |-  ( D  e.  Cat  ->  ( Id `  D )  Fn  ( Base `  D
) )
3128, 30syl 16 . . . . . . . . 9  |-  ( (
ph  /\  ( U  e.  ( F N G )  /\  V  e.  ( G N F ) ) )  -> 
( Id `  D
)  Fn  ( Base `  D ) )
32 dffn2 5525 . . . . . . . . 9  |-  ( ( Id `  D )  Fn  ( Base `  D
)  <->  ( Id `  D ) : (
Base `  D ) --> _V )
3331, 32sylib 189 . . . . . . . 8  |-  ( (
ph  /\  ( U  e.  ( F N G )  /\  V  e.  ( G N F ) ) )  -> 
( Id `  D
) : ( Base `  D ) --> _V )
34 relfunc 13979 . . . . . . . . . . 11  |-  Rel  ( C  Func  D )
35 1st2ndbr 6328 . . . . . . . . . . 11  |-  ( ( Rel  ( C  Func  D )  /\  F  e.  ( C  Func  D
) )  ->  ( 1st `  F ) ( C  Func  D )
( 2nd `  F
) )
3634, 8, 35sylancr 645 . . . . . . . . . 10  |-  ( ph  ->  ( 1st `  F
) ( C  Func  D ) ( 2nd `  F
) )
3720, 29, 36funcf1 13983 . . . . . . . . 9  |-  ( ph  ->  ( 1st `  F
) : B --> ( Base `  D ) )
3837adantr 452 . . . . . . . 8  |-  ( (
ph  /\  ( U  e.  ( F N G )  /\  V  e.  ( G N F ) ) )  -> 
( 1st `  F
) : B --> ( Base `  D ) )
39 fcompt 5836 . . . . . . . 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 643 . . . . . . 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 2412 . . . . . 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 2394 . . . . 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 2380 . . . . . . 7  |-  (  Hom  `  D )  =  (  Hom  `  D )
44 fucsect.t . . . . . . 7  |-  T  =  (Sect `  D )
4528adantr 452 . . . . . . 7  |-  ( ( ( ph  /\  ( U  e.  ( F N G )  /\  V  e.  ( G N F ) ) )  /\  x  e.  B )  ->  D  e.  Cat )
4638ffvelrnda 5802 . . . . . . 7  |-  ( ( ( ph  /\  ( U  e.  ( F N G )  /\  V  e.  ( G N F ) ) )  /\  x  e.  B )  ->  ( ( 1st `  F
) `  x )  e.  ( Base `  D
) )
47 1st2ndbr 6328 . . . . . . . . . . 11  |-  ( ( Rel  ( C  Func  D )  /\  G  e.  ( C  Func  D
) )  ->  ( 1st `  G ) ( C  Func  D )
( 2nd `  G
) )
4834, 14, 47sylancr 645 . . . . . . . . . 10  |-  ( ph  ->  ( 1st `  G
) ( C  Func  D ) ( 2nd `  G
) )
4920, 29, 48funcf1 13983 . . . . . . . . 9  |-  ( ph  ->  ( 1st `  G
) : B --> ( Base `  D ) )
5049adantr 452 . . . . . . . 8  |-  ( (
ph  /\  ( U  e.  ( F N G )  /\  V  e.  ( G N F ) ) )  -> 
( 1st `  G
) : B --> ( Base `  D ) )
5150ffvelrnda 5802 . . . . . . 7  |-  ( ( ( ph  /\  ( U  e.  ( F N G )  /\  V  e.  ( G N F ) ) )  /\  x  e.  B )  ->  ( ( 1st `  G
) `  x )  e.  ( Base `  D
) )
5222adantr 452 . . . . . . . . 9  |-  ( ( ( ph  /\  ( U  e.  ( F N G )  /\  V  e.  ( G N F ) ) )  /\  x  e.  B )  ->  U  e.  ( F N G ) )
533, 52nat1st2nd 14068 . . . . . . . 8  |-  ( ( ( 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 448 . . . . . . . 8  |-  ( ( ( ph  /\  ( U  e.  ( F N G )  /\  V  e.  ( G N F ) ) )  /\  x  e.  B )  ->  x  e.  B )
553, 53, 20, 43, 54natcl 14070 . . . . . . 7  |-  ( ( ( 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 452 . . . . . . . . 9  |-  ( ( ( ph  /\  ( U  e.  ( F N G )  /\  V  e.  ( G N F ) ) )  /\  x  e.  B )  ->  V  e.  ( G N F ) )
573, 56nat1st2nd 14068 . . . . . . . 8  |-  ( ( ( 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 14070 . . . . . . 7  |-  ( ( ( 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 13900 . . . . . 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 )
) ) )
6059ralbidva 2658 . . . . 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 )
) ) )
6119, 42, 603bitr4d 277 . . . 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
) ) )
6261pm5.32da 623 . . 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
) ) ) )
63 df-3an 938 . . 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 ) ) )
64 df-3an 938 . . 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
) ) )
6562, 63, 643bitr4g 280 . 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 ) ) ) )
6615, 65bitrd 245 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 177    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1717   A.wral 2642   _Vcvv 2892   <.cop 3753   class class class wbr 4146    e. cmpt 4200    o. ccom 4815   Rel wrel 4816    Fn wfn 5382   -->wf 5383   ` cfv 5387  (class class class)co 6013   1stc1st 6279   2ndc2nd 6280   Basecbs 13389    Hom chom 13460  compcco 13461   Catccat 13809   Idccid 13810  Sectcsect 13890    Func cfunc 13971   Nat cnat 14058   FuncCat cfuc 14059
This theorem is referenced by:  fucinv  14090
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361  ax-rep 4254  ax-sep 4264  ax-nul 4272  ax-pow 4311  ax-pr 4337  ax-un 4634  ax-cnex 8972  ax-resscn 8973  ax-1cn 8974  ax-icn 8975  ax-addcl 8976  ax-addrcl 8977  ax-mulcl 8978  ax-mulrcl 8979  ax-mulcom 8980  ax-addass 8981  ax-mulass 8982  ax-distr 8983  ax-i2m1 8984  ax-1ne0 8985  ax-1rid 8986  ax-rnegex 8987  ax-rrecex 8988  ax-cnre 8989  ax-pre-lttri 8990  ax-pre-lttrn 8991  ax-pre-ltadd 8992  ax-pre-mulgt0 8993
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2235  df-mo 2236  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ne 2545  df-nel 2546  df-ral 2647  df-rex 2648  df-reu 2649  df-rmo 2650  df-rab 2651  df-v 2894  df-sbc 3098  df-csb 3188  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-pss 3272  df-nul 3565  df-if 3676  df-pw 3737  df-sn 3756  df-pr 3757  df-tp 3758  df-op 3759  df-uni 3951  df-int 3986  df-iun 4030  df-br 4147  df-opab 4201  df-mpt 4202  df-tr 4237  df-eprel 4428  df-id 4432  df-po 4437  df-so 4438  df-fr 4475  df-we 4477  df-ord 4518  df-on 4519  df-lim 4520  df-suc 4521  df-om 4779  df-xp 4817  df-rel 4818  df-cnv 4819  df-co 4820  df-dm 4821  df-rn 4822  df-res 4823  df-ima 4824  df-iota 5351  df-fun 5389  df-fn 5390  df-f 5391  df-f1 5392  df-fo 5393  df-f1o 5394  df-fv 5395  df-ov 6016  df-oprab 6017  df-mpt2 6018  df-1st 6281  df-2nd 6282  df-riota 6478  df-recs 6562  df-rdg 6597  df-1o 6653  df-oadd 6657  df-er 6834  df-map 6949  df-ixp 6993  df-en 7039  df-dom 7040  df-sdom 7041  df-fin 7042  df-pnf 9048  df-mnf 9049  df-xr 9050  df-ltxr 9051  df-le 9052  df-sub 9218  df-neg 9219  df-nn 9926  df-2 9983  df-3 9984  df-4 9985  df-5 9986  df-6 9987  df-7 9988  df-8 9989  df-9 9990  df-10 9991  df-n0 10147  df-z 10208  df-dec 10308  df-uz 10414  df-fz 10969  df-struct 13391  df-ndx 13392  df-slot 13393  df-base 13394  df-hom 13473  df-cco 13474  df-cat 13813  df-cid 13814  df-sect 13893  df-func 13975  df-nat 14060  df-fuc 14061
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