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

Theorem setcinv 15936
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 2429 . . 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 15931 . . . 4  |-  ( U  e.  V  ->  C  e.  Cat )
63, 5syl 17 . . 3  |-  ( ph  ->  C  e.  Cat )
7 setcmon.x . . . 4  |-  ( ph  ->  X  e.  U )
84, 3setcbas 15924 . . . 4  |-  ( ph  ->  U  =  ( Base `  C ) )
97, 8eleqtrd 2519 . . 3  |-  ( ph  ->  X  e.  ( Base `  C ) )
10 setcmon.y . . . 4  |-  ( ph  ->  Y  e.  U )
1110, 8eleqtrd 2519 . . 3  |-  ( ph  ->  Y  e.  ( Base `  C ) )
12 eqid 2429 . . 3  |-  (Sect `  C )  =  (Sect `  C )
131, 2, 6, 9, 11, 12isinv 15616 . 2  |-  ( ph  ->  ( F ( X N Y ) G  <-> 
( F ( X (Sect `  C ) Y ) G  /\  G ( Y (Sect `  C ) X ) F ) ) )
144, 3, 7, 10, 12setcsect 15935 . . . . 5  |-  ( ph  ->  ( F ( X (Sect `  C ) Y ) G  <->  ( F : X --> Y  /\  G : Y --> X  /\  ( G  o.  F )  =  (  _I  |`  X ) ) ) )
15 df-3an 984 . . . . 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 264 . . . 4  |-  ( ph  ->  ( F ( X (Sect `  C ) Y ) G  <->  ( ( F : X --> Y  /\  G : Y --> X )  /\  ( G  o.  F )  =  (  _I  |`  X )
) ) )
174, 3, 10, 7, 12setcsect 15935 . . . . 5  |-  ( ph  ->  ( G ( Y (Sect `  C ) X ) F  <->  ( G : Y --> X  /\  F : X --> Y  /\  ( F  o.  G )  =  (  _I  |`  Y ) ) ) )
18 3ancoma 989 . . . . . 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 984 . . . . . 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 252 . . . . 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 264 . . . 4  |-  ( ph  ->  ( G ( Y (Sect `  C ) X ) F  <->  ( ( F : X --> Y  /\  G : Y --> X )  /\  ( F  o.  G )  =  (  _I  |`  Y )
) ) )
2216, 21anbi12d 715 . . 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 835 . . 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 266 . 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 6209 . . . . . 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 2438 . . . . . . 7  |-  ( `' F  =  G  <->  G  =  `' F )
2726anbi2i 698 . . . . . 6  |-  ( ( F : X -1-1-onto-> Y  /\  `' F  =  G
)  <->  ( F : X
-1-1-onto-> Y  /\  G  =  `' F ) )
2825, 27sylib 199 . . . . 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 809 . . . 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 467 . . 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 5831 . . . . 5  |-  ( F : X -1-1-onto-> Y  ->  F : X
--> Y )
3231ad2antrl 732 . . . 4  |-  ( (
ph  /\  ( F : X -1-1-onto-> Y  /\  G  =  `' F ) )  ->  F : X --> Y )
33 f1ocnv 5843 . . . . . . 7  |-  ( F : X -1-1-onto-> Y  ->  `' F : Y -1-1-onto-> X )
3433ad2antrl 732 . . . . . 6  |-  ( (
ph  /\  ( F : X -1-1-onto-> Y  /\  G  =  `' F ) )  ->  `' F : Y -1-1-onto-> X )
35 f1oeq1 5822 . . . . . . 7  |-  ( G  =  `' F  -> 
( G : Y -1-1-onto-> X  <->  `' F : Y -1-1-onto-> X ) )
3635ad2antll 733 . . . . . 6  |-  ( (
ph  /\  ( F : X -1-1-onto-> Y  /\  G  =  `' F ) )  -> 
( G : Y -1-1-onto-> X  <->  `' F : Y -1-1-onto-> X ) )
3734, 36mpbird 235 . . . . 5  |-  ( (
ph  /\  ( F : X -1-1-onto-> Y  /\  G  =  `' F ) )  ->  G : Y -1-1-onto-> X )
38 f1of 5831 . . . . 5  |-  ( G : Y -1-1-onto-> X  ->  G : Y
--> X )
3937, 38syl 17 . . . 4  |-  ( (
ph  /\  ( F : X -1-1-onto-> Y  /\  G  =  `' F ) )  ->  G : Y --> X )
40 simprr 764 . . . . . . 7  |-  ( (
ph  /\  ( F : X -1-1-onto-> Y  /\  G  =  `' F ) )  ->  G  =  `' F
)
4140coeq1d 5016 . . . . . 6  |-  ( (
ph  /\  ( F : X -1-1-onto-> Y  /\  G  =  `' F ) )  -> 
( G  o.  F
)  =  ( `' F  o.  F ) )
42 f1ococnv1 5859 . . . . . . 7  |-  ( F : X -1-1-onto-> Y  ->  ( `' F  o.  F )  =  (  _I  |`  X ) )
4342ad2antrl 732 . . . . . 6  |-  ( (
ph  /\  ( F : X -1-1-onto-> Y  /\  G  =  `' F ) )  -> 
( `' F  o.  F )  =  (  _I  |`  X )
)
4441, 43eqtrd 2470 . . . . 5  |-  ( (
ph  /\  ( F : X -1-1-onto-> Y  /\  G  =  `' F ) )  -> 
( G  o.  F
)  =  (  _I  |`  X ) )
4540coeq2d 5017 . . . . . 6  |-  ( (
ph  /\  ( F : X -1-1-onto-> Y  /\  G  =  `' F ) )  -> 
( F  o.  G
)  =  ( F  o.  `' F ) )
46 f1ococnv2 5857 . . . . . . 7  |-  ( F : X -1-1-onto-> Y  ->  ( F  o.  `' F )  =  (  _I  |`  Y )
)
4746ad2antrl 732 . . . . . 6  |-  ( (
ph  /\  ( F : X -1-1-onto-> Y  /\  G  =  `' F ) )  -> 
( F  o.  `' F )  =  (  _I  |`  Y )
)
4845, 47eqtrd 2470 . . . . 5  |-  ( (
ph  /\  ( F : X -1-1-onto-> Y  /\  G  =  `' F ) )  -> 
( F  o.  G
)  =  (  _I  |`  Y ) )
4944, 48jca 534 . . . 4  |-  ( (
ph  /\  ( F : X -1-1-onto-> Y  /\  G  =  `' F ) )  -> 
( ( G  o.  F )  =  (  _I  |`  X )  /\  ( F  o.  G
)  =  (  _I  |`  Y ) ) )
5032, 39, 49jca31 536 . . 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 840 . 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 282 1  |-  ( ph  ->  ( F ( X N Y ) G  <-> 
( F : X -1-1-onto-> Y  /\  G  =  `' F ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 187    /\ wa 370    /\ w3a 982    = wceq 1437    e. wcel 1870   class class class wbr 4426    _I cid 4764   `'ccnv 4853    |` cres 4856    o. ccom 4858   -->wf 5597   -1-1-onto->wf1o 5600   ` cfv 5601  (class class class)co 6305   Basecbs 15084   Catccat 15521  Sectcsect 15600  Invcinv 15601   SetCatcsetc 15921
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-rep 4538  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597  ax-cnex 9594  ax-resscn 9595  ax-1cn 9596  ax-icn 9597  ax-addcl 9598  ax-addrcl 9599  ax-mulcl 9600  ax-mulrcl 9601  ax-mulcom 9602  ax-addass 9603  ax-mulass 9604  ax-distr 9605  ax-i2m1 9606  ax-1ne0 9607  ax-1rid 9608  ax-rnegex 9609  ax-rrecex 9610  ax-cnre 9611  ax-pre-lttri 9612  ax-pre-lttrn 9613  ax-pre-ltadd 9614  ax-pre-mulgt0 9615
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-nel 2628  df-ral 2787  df-rex 2788  df-reu 2789  df-rmo 2790  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-pss 3458  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-tp 4007  df-op 4009  df-uni 4223  df-int 4259  df-iun 4304  df-br 4427  df-opab 4485  df-mpt 4486  df-tr 4521  df-eprel 4765  df-id 4769  df-po 4775  df-so 4776  df-fr 4813  df-we 4815  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-pred 5399  df-ord 5445  df-on 5446  df-lim 5447  df-suc 5448  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-riota 6267  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-om 6707  df-1st 6807  df-2nd 6808  df-wrecs 7036  df-recs 7098  df-rdg 7136  df-1o 7190  df-oadd 7194  df-er 7371  df-map 7482  df-en 7578  df-dom 7579  df-sdom 7580  df-fin 7581  df-pnf 9676  df-mnf 9677  df-xr 9678  df-ltxr 9679  df-le 9680  df-sub 9861  df-neg 9862  df-nn 10610  df-2 10668  df-3 10669  df-4 10670  df-5 10671  df-6 10672  df-7 10673  df-8 10674  df-9 10675  df-10 10676  df-n0 10870  df-z 10938  df-dec 11052  df-uz 11160  df-fz 11783  df-struct 15086  df-ndx 15087  df-slot 15088  df-base 15089  df-hom 15176  df-cco 15177  df-cat 15525  df-cid 15526  df-sect 15603  df-inv 15604  df-setc 15922
This theorem is referenced by:  setciso  15937  yonedainv  16117
  Copyright terms: Public domain W3C validator