MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  setcinv Unicode version

Theorem setcinv 13971
Description: An inverse in the category of sets is the converse operation. (Contributed by Mario Carneiro, 3-Jan-2017.)
Hypotheses
Ref Expression
setcmon.c  |-  C  =  ( SetCat `  U )
setcmon.u  |-  ( ph  ->  U  e.  V )
setcmon.x  |-  ( ph  ->  X  e.  U )
setcmon.y  |-  ( ph  ->  Y  e.  U )
setcinv.n  |-  N  =  (Inv `  C )
Assertion
Ref Expression
setcinv  |-  ( ph  ->  ( F ( X N Y ) G  <-> 
( F : X -1-1-onto-> Y  /\  G  =  `' F ) ) )

Proof of Theorem setcinv
StepHypRef Expression
1 eqid 2316 . . 3  |-  ( Base `  C )  =  (
Base `  C )
2 setcinv.n . . 3  |-  N  =  (Inv `  C )
3 setcmon.u . . . 4  |-  ( ph  ->  U  e.  V )
4 setcmon.c . . . . 5  |-  C  =  ( SetCat `  U )
54setccat 13966 . . . 4  |-  ( U  e.  V  ->  C  e.  Cat )
63, 5syl 15 . . 3  |-  ( ph  ->  C  e.  Cat )
7 setcmon.x . . . 4  |-  ( ph  ->  X  e.  U )
84, 3setcbas 13959 . . . 4  |-  ( ph  ->  U  =  ( Base `  C ) )
97, 8eleqtrd 2392 . . 3  |-  ( ph  ->  X  e.  ( Base `  C ) )
10 setcmon.y . . . 4  |-  ( ph  ->  Y  e.  U )
1110, 8eleqtrd 2392 . . 3  |-  ( ph  ->  Y  e.  ( Base `  C ) )
12 eqid 2316 . . 3  |-  (Sect `  C )  =  (Sect `  C )
131, 2, 6, 9, 11, 12isinv 13711 . 2  |-  ( ph  ->  ( F ( X N Y ) G  <-> 
( F ( X (Sect `  C ) Y ) G  /\  G ( Y (Sect `  C ) X ) F ) ) )
144, 3, 7, 10, 12setcsect 13970 . . . . 5  |-  ( ph  ->  ( F ( X (Sect `  C ) Y ) G  <->  ( F : X --> Y  /\  G : Y --> X  /\  ( G  o.  F )  =  (  _I  |`  X ) ) ) )
15 df-3an 936 . . . . 5  |-  ( ( F : X --> Y  /\  G : Y --> X  /\  ( G  o.  F
)  =  (  _I  |`  X ) )  <->  ( ( F : X --> Y  /\  G : Y --> X )  /\  ( G  o.  F )  =  (  _I  |`  X )
) )
1614, 15syl6bb 252 . . . 4  |-  ( ph  ->  ( F ( X (Sect `  C ) Y ) G  <->  ( ( F : X --> Y  /\  G : Y --> X )  /\  ( G  o.  F )  =  (  _I  |`  X )
) ) )
174, 3, 10, 7, 12setcsect 13970 . . . . 5  |-  ( ph  ->  ( G ( Y (Sect `  C ) X ) F  <->  ( G : Y --> X  /\  F : X --> Y  /\  ( F  o.  G )  =  (  _I  |`  Y ) ) ) )
18 3ancoma 941 . . . . . 6  |-  ( ( G : Y --> X  /\  F : X --> Y  /\  ( F  o.  G
)  =  (  _I  |`  Y ) )  <->  ( F : X --> Y  /\  G : Y --> X  /\  ( F  o.  G )  =  (  _I  |`  Y ) ) )
19 df-3an 936 . . . . . 6  |-  ( ( F : X --> Y  /\  G : Y --> X  /\  ( F  o.  G
)  =  (  _I  |`  Y ) )  <->  ( ( F : X --> Y  /\  G : Y --> X )  /\  ( F  o.  G )  =  (  _I  |`  Y )
) )
2018, 19bitri 240 . . . . 5  |-  ( ( G : Y --> X  /\  F : X --> Y  /\  ( F  o.  G
)  =  (  _I  |`  Y ) )  <->  ( ( F : X --> Y  /\  G : Y --> X )  /\  ( F  o.  G )  =  (  _I  |`  Y )
) )
2117, 20syl6bb 252 . . . 4  |-  ( ph  ->  ( G ( Y (Sect `  C ) X ) F  <->  ( ( F : X --> Y  /\  G : Y --> X )  /\  ( F  o.  G )  =  (  _I  |`  Y )
) ) )
2216, 21anbi12d 691 . . 3  |-  ( ph  ->  ( ( F ( X (Sect `  C
) Y ) G  /\  G ( Y (Sect `  C ) X ) F )  <-> 
( ( ( F : X --> Y  /\  G : Y --> X )  /\  ( G  o.  F )  =  (  _I  |`  X )
)  /\  ( ( F : X --> Y  /\  G : Y --> X )  /\  ( F  o.  G )  =  (  _I  |`  Y )
) ) ) )
23 anandi 801 . . 3  |-  ( ( ( F : X --> Y  /\  G : Y --> X )  /\  (
( G  o.  F
)  =  (  _I  |`  X )  /\  ( F  o.  G )  =  (  _I  |`  Y ) ) )  <->  ( (
( F : X --> Y  /\  G : Y --> X )  /\  ( G  o.  F )  =  (  _I  |`  X ) )  /\  ( ( F : X --> Y  /\  G : Y --> X )  /\  ( F  o.  G )  =  (  _I  |`  Y )
) ) )
2422, 23syl6bbr 254 . 2  |-  ( ph  ->  ( ( F ( X (Sect `  C
) Y ) G  /\  G ( Y (Sect `  C ) X ) F )  <-> 
( ( F : X
--> Y  /\  G : Y
--> X )  /\  (
( G  o.  F
)  =  (  _I  |`  X )  /\  ( F  o.  G )  =  (  _I  |`  Y ) ) ) ) )
25 fcof1o 5845 . . . . . 6  |-  ( ( ( F : X --> Y  /\  G : Y --> X )  /\  (
( F  o.  G
)  =  (  _I  |`  Y )  /\  ( G  o.  F )  =  (  _I  |`  X ) ) )  ->  ( F : X -1-1-onto-> Y  /\  `' F  =  G ) )
26 eqcom 2318 . . . . . . 7  |-  ( `' F  =  G  <->  G  =  `' F )
2726anbi2i 675 . . . . . 6  |-  ( ( F : X -1-1-onto-> Y  /\  `' F  =  G
)  <->  ( F : X
-1-1-onto-> Y  /\  G  =  `' F ) )
2825, 27sylib 188 . . . . 5  |-  ( ( ( F : X --> Y  /\  G : Y --> X )  /\  (
( F  o.  G
)  =  (  _I  |`  Y )  /\  ( G  o.  F )  =  (  _I  |`  X ) ) )  ->  ( F : X -1-1-onto-> Y  /\  G  =  `' F ) )
2928ancom2s 777 . . . 4  |-  ( ( ( F : X --> Y  /\  G : Y --> X )  /\  (
( G  o.  F
)  =  (  _I  |`  X )  /\  ( F  o.  G )  =  (  _I  |`  Y ) ) )  ->  ( F : X -1-1-onto-> Y  /\  G  =  `' F ) )
3029adantl 452 . . 3  |-  ( (
ph  /\  ( ( F : X --> Y  /\  G : Y --> X )  /\  ( ( G  o.  F )  =  (  _I  |`  X )  /\  ( F  o.  G )  =  (  _I  |`  Y )
) ) )  -> 
( F : X -1-1-onto-> Y  /\  G  =  `' F ) )
31 f1of 5510 . . . . 5  |-  ( F : X -1-1-onto-> Y  ->  F : X
--> Y )
3231ad2antrl 708 . . . 4  |-  ( (
ph  /\  ( F : X -1-1-onto-> Y  /\  G  =  `' F ) )  ->  F : X --> Y )
33 f1ocnv 5523 . . . . . . 7  |-  ( F : X -1-1-onto-> Y  ->  `' F : Y -1-1-onto-> X )
3433ad2antrl 708 . . . . . 6  |-  ( (
ph  /\  ( F : X -1-1-onto-> Y  /\  G  =  `' F ) )  ->  `' F : Y -1-1-onto-> X )
35 f1oeq1 5501 . . . . . . 7  |-  ( G  =  `' F  -> 
( G : Y -1-1-onto-> X  <->  `' F : Y -1-1-onto-> X ) )
3635ad2antll 709 . . . . . 6  |-  ( (
ph  /\  ( F : X -1-1-onto-> Y  /\  G  =  `' F ) )  -> 
( G : Y -1-1-onto-> X  <->  `' F : Y -1-1-onto-> X ) )
3734, 36mpbird 223 . . . . 5  |-  ( (
ph  /\  ( F : X -1-1-onto-> Y  /\  G  =  `' F ) )  ->  G : Y -1-1-onto-> X )
38 f1of 5510 . . . . 5  |-  ( G : Y -1-1-onto-> X  ->  G : Y
--> X )
3937, 38syl 15 . . . 4  |-  ( (
ph  /\  ( F : X -1-1-onto-> Y  /\  G  =  `' F ) )  ->  G : Y --> X )
40 simprr 733 . . . . . . 7  |-  ( (
ph  /\  ( F : X -1-1-onto-> Y  /\  G  =  `' F ) )  ->  G  =  `' F
)
4140coeq1d 4882 . . . . . 6  |-  ( (
ph  /\  ( F : X -1-1-onto-> Y  /\  G  =  `' F ) )  -> 
( G  o.  F
)  =  ( `' F  o.  F ) )
42 f1ococnv1 5540 . . . . . . 7  |-  ( F : X -1-1-onto-> Y  ->  ( `' F  o.  F )  =  (  _I  |`  X ) )
4342ad2antrl 708 . . . . . 6  |-  ( (
ph  /\  ( F : X -1-1-onto-> Y  /\  G  =  `' F ) )  -> 
( `' F  o.  F )  =  (  _I  |`  X )
)
4441, 43eqtrd 2348 . . . . 5  |-  ( (
ph  /\  ( F : X -1-1-onto-> Y  /\  G  =  `' F ) )  -> 
( G  o.  F
)  =  (  _I  |`  X ) )
4540coeq2d 4883 . . . . . 6  |-  ( (
ph  /\  ( F : X -1-1-onto-> Y  /\  G  =  `' F ) )  -> 
( F  o.  G
)  =  ( F  o.  `' F ) )
46 f1ococnv2 5538 . . . . . . 7  |-  ( F : X -1-1-onto-> Y  ->  ( F  o.  `' F )  =  (  _I  |`  Y )
)
4746ad2antrl 708 . . . . . 6  |-  ( (
ph  /\  ( F : X -1-1-onto-> Y  /\  G  =  `' F ) )  -> 
( F  o.  `' F )  =  (  _I  |`  Y )
)
4845, 47eqtrd 2348 . . . . 5  |-  ( (
ph  /\  ( F : X -1-1-onto-> Y  /\  G  =  `' F ) )  -> 
( F  o.  G
)  =  (  _I  |`  Y ) )
4944, 48jca 518 . . . 4  |-  ( (
ph  /\  ( F : X -1-1-onto-> Y  /\  G  =  `' F ) )  -> 
( ( G  o.  F )  =  (  _I  |`  X )  /\  ( F  o.  G
)  =  (  _I  |`  Y ) ) )
5032, 39, 49jca31 520 . . 3  |-  ( (
ph  /\  ( F : X -1-1-onto-> Y  /\  G  =  `' F ) )  -> 
( ( F : X
--> Y  /\  G : Y
--> X )  /\  (
( G  o.  F
)  =  (  _I  |`  X )  /\  ( F  o.  G )  =  (  _I  |`  Y ) ) ) )
5130, 50impbida 805 . 2  |-  ( ph  ->  ( ( ( F : X --> Y  /\  G : Y --> X )  /\  ( ( G  o.  F )  =  (  _I  |`  X )  /\  ( F  o.  G )  =  (  _I  |`  Y )
) )  <->  ( F : X -1-1-onto-> Y  /\  G  =  `' F ) ) )
5213, 24, 513bitrd 270 1  |-  ( ph  ->  ( F ( X N Y ) G  <-> 
( F : X -1-1-onto-> Y  /\  G  =  `' F ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1633    e. wcel 1701   class class class wbr 4060    _I cid 4341   `'ccnv 4725    |` cres 4728    o. ccom 4730   -->wf 5288   -1-1-onto->wf1o 5291   ` cfv 5292  (class class class)co 5900   Basecbs 13195   Catccat 13615  Sectcsect 13696  Invcinv 13697   SetCatcsetc 13956
This theorem is referenced by:  setciso  13972  yonedainv  14104
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-13 1703  ax-14 1705  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897  ax-ext 2297  ax-rep 4168  ax-sep 4178  ax-nul 4186  ax-pow 4225  ax-pr 4251  ax-un 4549  ax-cnex 8838  ax-resscn 8839  ax-1cn 8840  ax-icn 8841  ax-addcl 8842  ax-addrcl 8843  ax-mulcl 8844  ax-mulrcl 8845  ax-mulcom 8846  ax-addass 8847  ax-mulass 8848  ax-distr 8849  ax-i2m1 8850  ax-1ne0 8851  ax-1rid 8852  ax-rnegex 8853  ax-rrecex 8854  ax-cnre 8855  ax-pre-lttri 8856  ax-pre-lttrn 8857  ax-pre-ltadd 8858  ax-pre-mulgt0 8859
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 1533  df-nf 1536  df-sb 1640  df-eu 2180  df-mo 2181  df-clab 2303  df-cleq 2309  df-clel 2312  df-nfc 2441  df-ne 2481  df-nel 2482  df-ral 2582  df-rex 2583  df-reu 2584  df-rmo 2585  df-rab 2586  df-v 2824  df-sbc 3026  df-csb 3116  df-dif 3189  df-un 3191  df-in 3193  df-ss 3200  df-pss 3202  df-nul 3490  df-if 3600  df-pw 3661  df-sn 3680  df-pr 3681  df-tp 3682  df-op 3683  df-uni 3865  df-int 3900  df-iun 3944  df-br 4061  df-opab 4115  df-mpt 4116  df-tr 4151  df-eprel 4342  df-id 4346  df-po 4351  df-so 4352  df-fr 4389  df-we 4391  df-ord 4432  df-on 4433  df-lim 4434  df-suc 4435  df-om 4694  df-xp 4732  df-rel 4733  df-cnv 4734  df-co 4735  df-dm 4736  df-rn 4737  df-res 4738  df-ima 4739  df-iota 5256  df-fun 5294  df-fn 5295  df-f 5296  df-f1 5297  df-fo 5298  df-f1o 5299  df-fv 5300  df-ov 5903  df-oprab 5904  df-mpt2 5905  df-1st 6164  df-2nd 6165  df-riota 6346  df-recs 6430  df-rdg 6465  df-1o 6521  df-oadd 6525  df-er 6702  df-map 6817  df-en 6907  df-dom 6908  df-sdom 6909  df-fin 6910  df-pnf 8914  df-mnf 8915  df-xr 8916  df-ltxr 8917  df-le 8918  df-sub 9084  df-neg 9085  df-nn 9792  df-2 9849  df-3 9850  df-4 9851  df-5 9852  df-6 9853  df-7 9854  df-8 9855  df-9 9856  df-10 9857  df-n0 10013  df-z 10072  df-dec 10172  df-uz 10278  df-fz 10830  df-struct 13197  df-ndx 13198  df-slot 13199  df-base 13200  df-hom 13279  df-cco 13280  df-cat 13619  df-cid 13620  df-sect 13699  df-inv 13700  df-setc 13957
  Copyright terms: Public domain W3C validator