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Theorem fucinv 14171
Description: Two natural transformations are inverses of each other iff all the components are inverse. (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 ) )
fucinv.i  |-  I  =  (Inv `  Q )
fucinv.j  |-  J  =  (Inv `  D )
Assertion
Ref Expression
fucinv  |-  ( ph  ->  ( U ( F I G ) V  <-> 
( U  e.  ( F N G )  /\  V  e.  ( G N F )  /\  A. x  e.  B  ( U `  x ) ( ( ( 1st `  F
) `  x ) J ( ( 1st `  G ) `  x
) ) ( V `
 x ) ) ) )
Distinct variable groups:    x, B    x, C    x, D    x, I    x, F    x, G    x, J    x, N    x, V    ph, x    x, Q    x, U

Proof of Theorem fucinv
StepHypRef Expression
1 fuciso.q . . . 4  |-  Q  =  ( C FuncCat  D )
2 fuciso.b . . . 4  |-  B  =  ( Base `  C
)
3 fuciso.n . . . 4  |-  N  =  ( C Nat  D )
4 fuciso.f . . . 4  |-  ( ph  ->  F  e.  ( C 
Func  D ) )
5 fuciso.g . . . 4  |-  ( ph  ->  G  e.  ( C 
Func  D ) )
6 eqid 2437 . . . 4  |-  (Sect `  Q )  =  (Sect `  Q )
7 eqid 2437 . . . 4  |-  (Sect `  D )  =  (Sect `  D )
81, 2, 3, 4, 5, 6, 7fucsect 14170 . . 3  |-  ( ph  ->  ( U ( F (Sect `  Q ) G ) V  <->  ( U  e.  ( F N G )  /\  V  e.  ( G N F )  /\  A. x  e.  B  ( U `  x ) ( ( ( 1st `  F
) `  x )
(Sect `  D )
( ( 1st `  G
) `  x )
) ( V `  x ) ) ) )
91, 2, 3, 5, 4, 6, 7fucsect 14170 . . 3  |-  ( ph  ->  ( V ( G (Sect `  Q ) F ) U  <->  ( V  e.  ( G N F )  /\  U  e.  ( F N G )  /\  A. x  e.  B  ( V `  x ) ( ( ( 1st `  G
) `  x )
(Sect `  D )
( ( 1st `  F
) `  x )
) ( U `  x ) ) ) )
108, 9anbi12d 693 . 2  |-  ( ph  ->  ( ( U ( F (Sect `  Q
) G ) V  /\  V ( G (Sect `  Q ) F ) U )  <-> 
( ( U  e.  ( F N G )  /\  V  e.  ( G N F )  /\  A. x  e.  B  ( U `  x ) ( ( ( 1st `  F
) `  x )
(Sect `  D )
( ( 1st `  G
) `  x )
) ( V `  x ) )  /\  ( V  e.  ( G N F )  /\  U  e.  ( F N G )  /\  A. x  e.  B  ( V `  x )
( ( ( 1st `  G ) `  x
) (Sect `  D
) ( ( 1st `  F ) `  x
) ) ( U `
 x ) ) ) ) )
111fucbas 14158 . . 3  |-  ( C 
Func  D )  =  (
Base `  Q )
12 fucinv.i . . 3  |-  I  =  (Inv `  Q )
13 funcrcl 14061 . . . . . 6  |-  ( F  e.  ( C  Func  D )  ->  ( C  e.  Cat  /\  D  e. 
Cat ) )
144, 13syl 16 . . . . 5  |-  ( ph  ->  ( C  e.  Cat  /\  D  e.  Cat )
)
1514simpld 447 . . . 4  |-  ( ph  ->  C  e.  Cat )
1614simprd 451 . . . 4  |-  ( ph  ->  D  e.  Cat )
171, 15, 16fuccat 14168 . . 3  |-  ( ph  ->  Q  e.  Cat )
1811, 12, 17, 4, 5, 6isinv 13986 . 2  |-  ( ph  ->  ( U ( F I G ) V  <-> 
( U ( F (Sect `  Q ) G ) V  /\  V ( G (Sect `  Q ) F ) U ) ) )
19 eqid 2437 . . . . . . 7  |-  ( Base `  D )  =  (
Base `  D )
20 fucinv.j . . . . . . 7  |-  J  =  (Inv `  D )
2116adantr 453 . . . . . . 7  |-  ( (
ph  /\  x  e.  B )  ->  D  e.  Cat )
22 relfunc 14060 . . . . . . . . . 10  |-  Rel  ( C  Func  D )
23 1st2ndbr 6397 . . . . . . . . . 10  |-  ( ( Rel  ( C  Func  D )  /\  F  e.  ( C  Func  D
) )  ->  ( 1st `  F ) ( C  Func  D )
( 2nd `  F
) )
2422, 4, 23sylancr 646 . . . . . . . . 9  |-  ( ph  ->  ( 1st `  F
) ( C  Func  D ) ( 2nd `  F
) )
252, 19, 24funcf1 14064 . . . . . . . 8  |-  ( ph  ->  ( 1st `  F
) : B --> ( Base `  D ) )
2625ffvelrnda 5871 . . . . . . 7  |-  ( (
ph  /\  x  e.  B )  ->  (
( 1st `  F
) `  x )  e.  ( Base `  D
) )
27 1st2ndbr 6397 . . . . . . . . . 10  |-  ( ( Rel  ( C  Func  D )  /\  G  e.  ( C  Func  D
) )  ->  ( 1st `  G ) ( C  Func  D )
( 2nd `  G
) )
2822, 5, 27sylancr 646 . . . . . . . . 9  |-  ( ph  ->  ( 1st `  G
) ( C  Func  D ) ( 2nd `  G
) )
292, 19, 28funcf1 14064 . . . . . . . 8  |-  ( ph  ->  ( 1st `  G
) : B --> ( Base `  D ) )
3029ffvelrnda 5871 . . . . . . 7  |-  ( (
ph  /\  x  e.  B )  ->  (
( 1st `  G
) `  x )  e.  ( Base `  D
) )
3119, 20, 21, 26, 30, 7isinv 13986 . . . . . 6  |-  ( (
ph  /\  x  e.  B )  ->  (
( U `  x
) ( ( ( 1st `  F ) `
 x ) J ( ( 1st `  G
) `  x )
) ( V `  x )  <->  ( ( U `  x )
( ( ( 1st `  F ) `  x
) (Sect `  D
) ( ( 1st `  G ) `  x
) ) ( V `
 x )  /\  ( V `  x ) ( ( ( 1st `  G ) `  x
) (Sect `  D
) ( ( 1st `  F ) `  x
) ) ( U `
 x ) ) ) )
3231ralbidva 2722 . . . . 5  |-  ( ph  ->  ( A. x  e.  B  ( U `  x ) ( ( ( 1st `  F
) `  x ) J ( ( 1st `  G ) `  x
) ) ( V `
 x )  <->  A. x  e.  B  ( ( U `  x )
( ( ( 1st `  F ) `  x
) (Sect `  D
) ( ( 1st `  G ) `  x
) ) ( V `
 x )  /\  ( V `  x ) ( ( ( 1st `  G ) `  x
) (Sect `  D
) ( ( 1st `  F ) `  x
) ) ( U `
 x ) ) ) )
33 r19.26 2839 . . . . 5  |-  ( A. x  e.  B  (
( U `  x
) ( ( ( 1st `  F ) `
 x ) (Sect `  D ) ( ( 1st `  G ) `
 x ) ) ( V `  x
)  /\  ( V `  x ) ( ( ( 1st `  G
) `  x )
(Sect `  D )
( ( 1st `  F
) `  x )
) ( U `  x ) )  <->  ( A. x  e.  B  ( U `  x )
( ( ( 1st `  F ) `  x
) (Sect `  D
) ( ( 1st `  G ) `  x
) ) ( V `
 x )  /\  A. x  e.  B  ( V `  x ) ( ( ( 1st `  G ) `  x
) (Sect `  D
) ( ( 1st `  F ) `  x
) ) ( U `
 x ) ) )
3432, 33syl6bb 254 . . . 4  |-  ( ph  ->  ( A. x  e.  B  ( U `  x ) ( ( ( 1st `  F
) `  x ) J ( ( 1st `  G ) `  x
) ) ( V `
 x )  <->  ( A. x  e.  B  ( U `  x )
( ( ( 1st `  F ) `  x
) (Sect `  D
) ( ( 1st `  G ) `  x
) ) ( V `
 x )  /\  A. x  e.  B  ( V `  x ) ( ( ( 1st `  G ) `  x
) (Sect `  D
) ( ( 1st `  F ) `  x
) ) ( U `
 x ) ) ) )
3534anbi2d 686 . . 3  |-  ( ph  ->  ( ( ( U  e.  ( F N G )  /\  V  e.  ( G N F ) )  /\  A. x  e.  B  ( U `  x )
( ( ( 1st `  F ) `  x
) J ( ( 1st `  G ) `
 x ) ) ( V `  x
) )  <->  ( ( U  e.  ( F N G )  /\  V  e.  ( G N F ) )  /\  ( A. x  e.  B  ( U `  x ) ( ( ( 1st `  F ) `  x
) (Sect `  D
) ( ( 1st `  G ) `  x
) ) ( V `
 x )  /\  A. x  e.  B  ( V `  x ) ( ( ( 1st `  G ) `  x
) (Sect `  D
) ( ( 1st `  F ) `  x
) ) ( U `
 x ) ) ) ) )
36 df-3an 939 . . 3  |-  ( ( U  e.  ( F N G )  /\  V  e.  ( G N F )  /\  A. x  e.  B  ( U `  x )
( ( ( 1st `  F ) `  x
) J ( ( 1st `  G ) `
 x ) ) ( V `  x
) )  <->  ( ( U  e.  ( F N G )  /\  V  e.  ( G N F ) )  /\  A. x  e.  B  ( U `  x )
( ( ( 1st `  F ) `  x
) J ( ( 1st `  G ) `
 x ) ) ( V `  x
) ) )
37 df-3an 939 . . . . 5  |-  ( ( U  e.  ( F N G )  /\  V  e.  ( G N F )  /\  A. x  e.  B  ( U `  x )
( ( ( 1st `  F ) `  x
) (Sect `  D
) ( ( 1st `  G ) `  x
) ) ( V `
 x ) )  <-> 
( ( U  e.  ( F N G )  /\  V  e.  ( G N F ) )  /\  A. x  e.  B  ( U `  x )
( ( ( 1st `  F ) `  x
) (Sect `  D
) ( ( 1st `  G ) `  x
) ) ( V `
 x ) ) )
38 3ancoma 944 . . . . . 6  |-  ( ( V  e.  ( G N F )  /\  U  e.  ( F N G )  /\  A. x  e.  B  ( V `  x )
( ( ( 1st `  G ) `  x
) (Sect `  D
) ( ( 1st `  F ) `  x
) ) ( U `
 x ) )  <-> 
( U  e.  ( F N G )  /\  V  e.  ( G N F )  /\  A. x  e.  B  ( V `  x ) ( ( ( 1st `  G
) `  x )
(Sect `  D )
( ( 1st `  F
) `  x )
) ( U `  x ) ) )
39 df-3an 939 . . . . . 6  |-  ( ( U  e.  ( F N G )  /\  V  e.  ( G N F )  /\  A. x  e.  B  ( V `  x )
( ( ( 1st `  G ) `  x
) (Sect `  D
) ( ( 1st `  F ) `  x
) ) ( U `
 x ) )  <-> 
( ( U  e.  ( F N G )  /\  V  e.  ( G N F ) )  /\  A. x  e.  B  ( V `  x )
( ( ( 1st `  G ) `  x
) (Sect `  D
) ( ( 1st `  F ) `  x
) ) ( U `
 x ) ) )
4038, 39bitri 242 . . . . 5  |-  ( ( V  e.  ( G N F )  /\  U  e.  ( F N G )  /\  A. x  e.  B  ( V `  x )
( ( ( 1st `  G ) `  x
) (Sect `  D
) ( ( 1st `  F ) `  x
) ) ( U `
 x ) )  <-> 
( ( U  e.  ( F N G )  /\  V  e.  ( G N F ) )  /\  A. x  e.  B  ( V `  x )
( ( ( 1st `  G ) `  x
) (Sect `  D
) ( ( 1st `  F ) `  x
) ) ( U `
 x ) ) )
4137, 40anbi12i 680 . . . 4  |-  ( ( ( U  e.  ( F N G )  /\  V  e.  ( G N F )  /\  A. x  e.  B  ( U `  x ) ( ( ( 1st `  F
) `  x )
(Sect `  D )
( ( 1st `  G
) `  x )
) ( V `  x ) )  /\  ( V  e.  ( G N F )  /\  U  e.  ( F N G )  /\  A. x  e.  B  ( V `  x )
( ( ( 1st `  G ) `  x
) (Sect `  D
) ( ( 1st `  F ) `  x
) ) ( U `
 x ) ) )  <->  ( ( ( U  e.  ( F N G )  /\  V  e.  ( G N F ) )  /\  A. x  e.  B  ( U `  x ) ( ( ( 1st `  F ) `  x
) (Sect `  D
) ( ( 1st `  G ) `  x
) ) ( V `
 x ) )  /\  ( ( U  e.  ( F N G )  /\  V  e.  ( G N F ) )  /\  A. x  e.  B  ( V `  x )
( ( ( 1st `  G ) `  x
) (Sect `  D
) ( ( 1st `  F ) `  x
) ) ( U `
 x ) ) ) )
42 anandi 803 . . . 4  |-  ( ( ( U  e.  ( F N G )  /\  V  e.  ( G N F ) )  /\  ( A. x  e.  B  ( U `  x )
( ( ( 1st `  F ) `  x
) (Sect `  D
) ( ( 1st `  G ) `  x
) ) ( V `
 x )  /\  A. x  e.  B  ( V `  x ) ( ( ( 1st `  G ) `  x
) (Sect `  D
) ( ( 1st `  F ) `  x
) ) ( U `
 x ) ) )  <->  ( ( ( U  e.  ( F N G )  /\  V  e.  ( G N F ) )  /\  A. x  e.  B  ( U `  x ) ( ( ( 1st `  F ) `  x
) (Sect `  D
) ( ( 1st `  G ) `  x
) ) ( V `
 x ) )  /\  ( ( U  e.  ( F N G )  /\  V  e.  ( G N F ) )  /\  A. x  e.  B  ( V `  x )
( ( ( 1st `  G ) `  x
) (Sect `  D
) ( ( 1st `  F ) `  x
) ) ( U `
 x ) ) ) )
4341, 42bitr4i 245 . . 3  |-  ( ( ( U  e.  ( F N G )  /\  V  e.  ( G N F )  /\  A. x  e.  B  ( U `  x ) ( ( ( 1st `  F
) `  x )
(Sect `  D )
( ( 1st `  G
) `  x )
) ( V `  x ) )  /\  ( V  e.  ( G N F )  /\  U  e.  ( F N G )  /\  A. x  e.  B  ( V `  x )
( ( ( 1st `  G ) `  x
) (Sect `  D
) ( ( 1st `  F ) `  x
) ) ( U `
 x ) ) )  <->  ( ( U  e.  ( F N G )  /\  V  e.  ( G N F ) )  /\  ( A. x  e.  B  ( U `  x ) ( ( ( 1st `  F ) `  x
) (Sect `  D
) ( ( 1st `  G ) `  x
) ) ( V `
 x )  /\  A. x  e.  B  ( V `  x ) ( ( ( 1st `  G ) `  x
) (Sect `  D
) ( ( 1st `  F ) `  x
) ) ( U `
 x ) ) ) )
4435, 36, 433bitr4g 281 . 2  |-  ( ph  ->  ( ( U  e.  ( F N G )  /\  V  e.  ( G N F )  /\  A. x  e.  B  ( U `  x ) ( ( ( 1st `  F
) `  x ) J ( ( 1st `  G ) `  x
) ) ( V `
 x ) )  <-> 
( ( U  e.  ( F N G )  /\  V  e.  ( G N F )  /\  A. x  e.  B  ( U `  x ) ( ( ( 1st `  F
) `  x )
(Sect `  D )
( ( 1st `  G
) `  x )
) ( V `  x ) )  /\  ( V  e.  ( G N F )  /\  U  e.  ( F N G )  /\  A. x  e.  B  ( V `  x )
( ( ( 1st `  G ) `  x
) (Sect `  D
) ( ( 1st `  F ) `  x
) ) ( U `
 x ) ) ) ) )
4510, 18, 443bitr4d 278 1  |-  ( ph  ->  ( U ( F I G ) V  <-> 
( U  e.  ( F N G )  /\  V  e.  ( G N F )  /\  A. x  e.  B  ( U `  x ) ( ( ( 1st `  F
) `  x ) J ( ( 1st `  G ) `  x
) ) ( V `
 x ) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1726   A.wral 2706   class class class wbr 4213   Rel wrel 4884   ` cfv 5455  (class class class)co 6082   1stc1st 6348   2ndc2nd 6349   Basecbs 13470   Catccat 13890  Sectcsect 13971  Invcinv 13972    Func cfunc 14052   Nat cnat 14139   FuncCat cfuc 14140
This theorem is referenced by:  invfuc  14172  fuciso  14173
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2418  ax-rep 4321  ax-sep 4331  ax-nul 4339  ax-pow 4378  ax-pr 4404  ax-un 4702  ax-cnex 9047  ax-resscn 9048  ax-1cn 9049  ax-icn 9050  ax-addcl 9051  ax-addrcl 9052  ax-mulcl 9053  ax-mulrcl 9054  ax-mulcom 9055  ax-addass 9056  ax-mulass 9057  ax-distr 9058  ax-i2m1 9059  ax-1ne0 9060  ax-1rid 9061  ax-rnegex 9062  ax-rrecex 9063  ax-cnre 9064  ax-pre-lttri 9065  ax-pre-lttrn 9066  ax-pre-ltadd 9067  ax-pre-mulgt0 9068
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2286  df-mo 2287  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ne 2602  df-nel 2603  df-ral 2711  df-rex 2712  df-reu 2713  df-rmo 2714  df-rab 2715  df-v 2959  df-sbc 3163  df-csb 3253  df-dif 3324  df-un 3326  df-in 3328  df-ss 3335  df-pss 3337  df-nul 3630  df-if 3741  df-pw 3802  df-sn 3821  df-pr 3822  df-tp 3823  df-op 3824  df-uni 4017  df-int 4052  df-iun 4096  df-br 4214  df-opab 4268  df-mpt 4269  df-tr 4304  df-eprel 4495  df-id 4499  df-po 4504  df-so 4505  df-fr 4542  df-we 4544  df-ord 4585  df-on 4586  df-lim 4587  df-suc 4588  df-om 4847  df-xp 4885  df-rel 4886  df-cnv 4887  df-co 4888  df-dm 4889  df-rn 4890  df-res 4891  df-ima 4892  df-iota 5419  df-fun 5457  df-fn 5458  df-f 5459  df-f1 5460  df-fo 5461  df-f1o 5462  df-fv 5463  df-ov 6085  df-oprab 6086  df-mpt2 6087  df-1st 6350  df-2nd 6351  df-riota 6550  df-recs 6634  df-rdg 6669  df-1o 6725  df-oadd 6729  df-er 6906  df-map 7021  df-ixp 7065  df-en 7111  df-dom 7112  df-sdom 7113  df-fin 7114  df-pnf 9123  df-mnf 9124  df-xr 9125  df-ltxr 9126  df-le 9127  df-sub 9294  df-neg 9295  df-nn 10002  df-2 10059  df-3 10060  df-4 10061  df-5 10062  df-6 10063  df-7 10064  df-8 10065  df-9 10066  df-10 10067  df-n0 10223  df-z 10284  df-dec 10384  df-uz 10490  df-fz 11045  df-struct 13472  df-ndx 13473  df-slot 13474  df-base 13475  df-hom 13554  df-cco 13555  df-cat 13894  df-cid 13895  df-sect 13974  df-inv 13975  df-func 14056  df-nat 14141  df-fuc 14142
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