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Theorem setcinv 15278
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 2467 . . 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 15273 . . . 4  |-  ( U  e.  V  ->  C  e.  Cat )
63, 5syl 16 . . 3  |-  ( ph  ->  C  e.  Cat )
7 setcmon.x . . . 4  |-  ( ph  ->  X  e.  U )
84, 3setcbas 15266 . . . 4  |-  ( ph  ->  U  =  ( Base `  C ) )
97, 8eleqtrd 2557 . . 3  |-  ( ph  ->  X  e.  ( Base `  C ) )
10 setcmon.y . . . 4  |-  ( ph  ->  Y  e.  U )
1110, 8eleqtrd 2557 . . 3  |-  ( ph  ->  Y  e.  ( Base `  C ) )
12 eqid 2467 . . 3  |-  (Sect `  C )  =  (Sect `  C )
131, 2, 6, 9, 11, 12isinv 15018 . 2  |-  ( ph  ->  ( F ( X N Y ) G  <-> 
( F ( X (Sect `  C ) Y ) G  /\  G ( Y (Sect `  C ) X ) F ) ) )
144, 3, 7, 10, 12setcsect 15277 . . . . 5  |-  ( ph  ->  ( F ( X (Sect `  C ) Y ) G  <->  ( F : X --> Y  /\  G : Y --> X  /\  ( G  o.  F )  =  (  _I  |`  X ) ) ) )
15 df-3an 975 . . . . 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 261 . . . 4  |-  ( ph  ->  ( F ( X (Sect `  C ) Y ) G  <->  ( ( F : X --> Y  /\  G : Y --> X )  /\  ( G  o.  F )  =  (  _I  |`  X )
) ) )
174, 3, 10, 7, 12setcsect 15277 . . . . 5  |-  ( ph  ->  ( G ( Y (Sect `  C ) X ) F  <->  ( G : Y --> X  /\  F : X --> Y  /\  ( F  o.  G )  =  (  _I  |`  Y ) ) ) )
18 3ancoma 980 . . . . . 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 975 . . . . . 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 249 . . . . 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 261 . . . 4  |-  ( ph  ->  ( G ( Y (Sect `  C ) X ) F  <->  ( ( F : X --> Y  /\  G : Y --> X )  /\  ( F  o.  G )  =  (  _I  |`  Y )
) ) )
2216, 21anbi12d 710 . . 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 826 . . 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 263 . 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 6188 . . . . . 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 2476 . . . . . . 7  |-  ( `' F  =  G  <->  G  =  `' F )
2726anbi2i 694 . . . . . 6  |-  ( ( F : X -1-1-onto-> Y  /\  `' F  =  G
)  <->  ( F : X
-1-1-onto-> Y  /\  G  =  `' F ) )
2825, 27sylib 196 . . . . 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 800 . . . 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 466 . . 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 5816 . . . . 5  |-  ( F : X -1-1-onto-> Y  ->  F : X
--> Y )
3231ad2antrl 727 . . . 4  |-  ( (
ph  /\  ( F : X -1-1-onto-> Y  /\  G  =  `' F ) )  ->  F : X --> Y )
33 f1ocnv 5828 . . . . . . 7  |-  ( F : X -1-1-onto-> Y  ->  `' F : Y -1-1-onto-> X )
3433ad2antrl 727 . . . . . 6  |-  ( (
ph  /\  ( F : X -1-1-onto-> Y  /\  G  =  `' F ) )  ->  `' F : Y -1-1-onto-> X )
35 f1oeq1 5807 . . . . . . 7  |-  ( G  =  `' F  -> 
( G : Y -1-1-onto-> X  <->  `' F : Y -1-1-onto-> X ) )
3635ad2antll 728 . . . . . 6  |-  ( (
ph  /\  ( F : X -1-1-onto-> Y  /\  G  =  `' F ) )  -> 
( G : Y -1-1-onto-> X  <->  `' F : Y -1-1-onto-> X ) )
3734, 36mpbird 232 . . . . 5  |-  ( (
ph  /\  ( F : X -1-1-onto-> Y  /\  G  =  `' F ) )  ->  G : Y -1-1-onto-> X )
38 f1of 5816 . . . . 5  |-  ( G : Y -1-1-onto-> X  ->  G : Y
--> X )
3937, 38syl 16 . . . 4  |-  ( (
ph  /\  ( F : X -1-1-onto-> Y  /\  G  =  `' F ) )  ->  G : Y --> X )
40 simprr 756 . . . . . . 7  |-  ( (
ph  /\  ( F : X -1-1-onto-> Y  /\  G  =  `' F ) )  ->  G  =  `' F
)
4140coeq1d 5164 . . . . . 6  |-  ( (
ph  /\  ( F : X -1-1-onto-> Y  /\  G  =  `' F ) )  -> 
( G  o.  F
)  =  ( `' F  o.  F ) )
42 f1ococnv1 5844 . . . . . . 7  |-  ( F : X -1-1-onto-> Y  ->  ( `' F  o.  F )  =  (  _I  |`  X ) )
4342ad2antrl 727 . . . . . 6  |-  ( (
ph  /\  ( F : X -1-1-onto-> Y  /\  G  =  `' F ) )  -> 
( `' F  o.  F )  =  (  _I  |`  X )
)
4441, 43eqtrd 2508 . . . . 5  |-  ( (
ph  /\  ( F : X -1-1-onto-> Y  /\  G  =  `' F ) )  -> 
( G  o.  F
)  =  (  _I  |`  X ) )
4540coeq2d 5165 . . . . . 6  |-  ( (
ph  /\  ( F : X -1-1-onto-> Y  /\  G  =  `' F ) )  -> 
( F  o.  G
)  =  ( F  o.  `' F ) )
46 f1ococnv2 5842 . . . . . . 7  |-  ( F : X -1-1-onto-> Y  ->  ( F  o.  `' F )  =  (  _I  |`  Y )
)
4746ad2antrl 727 . . . . . 6  |-  ( (
ph  /\  ( F : X -1-1-onto-> Y  /\  G  =  `' F ) )  -> 
( F  o.  `' F )  =  (  _I  |`  Y )
)
4845, 47eqtrd 2508 . . . . 5  |-  ( (
ph  /\  ( F : X -1-1-onto-> Y  /\  G  =  `' F ) )  -> 
( F  o.  G
)  =  (  _I  |`  Y ) )
4944, 48jca 532 . . . 4  |-  ( (
ph  /\  ( F : X -1-1-onto-> Y  /\  G  =  `' F ) )  -> 
( ( G  o.  F )  =  (  _I  |`  X )  /\  ( F  o.  G
)  =  (  _I  |`  Y ) ) )
5032, 39, 49jca31 534 . . 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 830 . 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 279 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 184    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767   class class class wbr 4447    _I cid 4790   `'ccnv 4998    |` cres 5001    o. ccom 5003   -->wf 5584   -1-1-onto->wf1o 5587   ` cfv 5588  (class class class)co 6285   Basecbs 14493   Catccat 14922  Sectcsect 15003  Invcinv 15004   SetCatcsetc 15263
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6577  ax-cnex 9549  ax-resscn 9550  ax-1cn 9551  ax-icn 9552  ax-addcl 9553  ax-addrcl 9554  ax-mulcl 9555  ax-mulrcl 9556  ax-mulcom 9557  ax-addass 9558  ax-mulass 9559  ax-distr 9560  ax-i2m1 9561  ax-1ne0 9562  ax-1rid 9563  ax-rnegex 9564  ax-rrecex 9565  ax-cnre 9566  ax-pre-lttri 9567  ax-pre-lttrn 9568  ax-pre-ltadd 9569  ax-pre-mulgt0 9570
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-riota 6246  df-ov 6288  df-oprab 6289  df-mpt2 6290  df-om 6686  df-1st 6785  df-2nd 6786  df-recs 7043  df-rdg 7077  df-1o 7131  df-oadd 7135  df-er 7312  df-map 7423  df-en 7518  df-dom 7519  df-sdom 7520  df-fin 7521  df-pnf 9631  df-mnf 9632  df-xr 9633  df-ltxr 9634  df-le 9635  df-sub 9808  df-neg 9809  df-nn 10538  df-2 10595  df-3 10596  df-4 10597  df-5 10598  df-6 10599  df-7 10600  df-8 10601  df-9 10602  df-10 10603  df-n0 10797  df-z 10866  df-dec 10978  df-uz 11084  df-fz 11674  df-struct 14495  df-ndx 14496  df-slot 14497  df-base 14498  df-hom 14582  df-cco 14583  df-cat 14926  df-cid 14927  df-sect 15006  df-inv 15007  df-setc 15264
This theorem is referenced by:  setciso  15279  yonedainv  15411
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