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Theorem setcinv 14233
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 2435 . . 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 14228 . . . 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 14221 . . . 4  |-  ( ph  ->  U  =  ( Base `  C ) )
97, 8eleqtrd 2511 . . 3  |-  ( ph  ->  X  e.  ( Base `  C ) )
10 setcmon.y . . . 4  |-  ( ph  ->  Y  e.  U )
1110, 8eleqtrd 2511 . . 3  |-  ( ph  ->  Y  e.  ( Base `  C ) )
12 eqid 2435 . . 3  |-  (Sect `  C )  =  (Sect `  C )
131, 2, 6, 9, 11, 12isinv 13973 . 2  |-  ( ph  ->  ( F ( X N Y ) G  <-> 
( F ( X (Sect `  C ) Y ) G  /\  G ( Y (Sect `  C ) X ) F ) ) )
144, 3, 7, 10, 12setcsect 14232 . . . . 5  |-  ( ph  ->  ( F ( X (Sect `  C ) Y ) G  <->  ( F : X --> Y  /\  G : Y --> X  /\  ( G  o.  F )  =  (  _I  |`  X ) ) ) )
15 df-3an 938 . . . . 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 253 . . . 4  |-  ( ph  ->  ( F ( X (Sect `  C ) Y ) G  <->  ( ( F : X --> Y  /\  G : Y --> X )  /\  ( G  o.  F )  =  (  _I  |`  X )
) ) )
174, 3, 10, 7, 12setcsect 14232 . . . . 5  |-  ( ph  ->  ( G ( Y (Sect `  C ) X ) F  <->  ( G : Y --> X  /\  F : X --> Y  /\  ( F  o.  G )  =  (  _I  |`  Y ) ) ) )
18 3ancoma 943 . . . . . 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 938 . . . . . 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 241 . . . . 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 253 . . . 4  |-  ( ph  ->  ( G ( Y (Sect `  C ) X ) F  <->  ( ( F : X --> Y  /\  G : Y --> X )  /\  ( F  o.  G )  =  (  _I  |`  Y )
) ) )
2216, 21anbi12d 692 . . 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 802 . . 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 255 . 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 6017 . . . . . 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 2437 . . . . . . 7  |-  ( `' F  =  G  <->  G  =  `' F )
2726anbi2i 676 . . . . . 6  |-  ( ( F : X -1-1-onto-> Y  /\  `' F  =  G
)  <->  ( F : X
-1-1-onto-> Y  /\  G  =  `' F ) )
2825, 27sylib 189 . . . . 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 778 . . . 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 453 . . 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 5665 . . . . 5  |-  ( F : X -1-1-onto-> Y  ->  F : X
--> Y )
3231ad2antrl 709 . . . 4  |-  ( (
ph  /\  ( F : X -1-1-onto-> Y  /\  G  =  `' F ) )  ->  F : X --> Y )
33 f1ocnv 5678 . . . . . . 7  |-  ( F : X -1-1-onto-> Y  ->  `' F : Y -1-1-onto-> X )
3433ad2antrl 709 . . . . . 6  |-  ( (
ph  /\  ( F : X -1-1-onto-> Y  /\  G  =  `' F ) )  ->  `' F : Y -1-1-onto-> X )
35 f1oeq1 5656 . . . . . . 7  |-  ( G  =  `' F  -> 
( G : Y -1-1-onto-> X  <->  `' F : Y -1-1-onto-> X ) )
3635ad2antll 710 . . . . . 6  |-  ( (
ph  /\  ( F : X -1-1-onto-> Y  /\  G  =  `' F ) )  -> 
( G : Y -1-1-onto-> X  <->  `' F : Y -1-1-onto-> X ) )
3734, 36mpbird 224 . . . . 5  |-  ( (
ph  /\  ( F : X -1-1-onto-> Y  /\  G  =  `' F ) )  ->  G : Y -1-1-onto-> X )
38 f1of 5665 . . . . 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 734 . . . . . . 7  |-  ( (
ph  /\  ( F : X -1-1-onto-> Y  /\  G  =  `' F ) )  ->  G  =  `' F
)
4140coeq1d 5025 . . . . . 6  |-  ( (
ph  /\  ( F : X -1-1-onto-> Y  /\  G  =  `' F ) )  -> 
( G  o.  F
)  =  ( `' F  o.  F ) )
42 f1ococnv1 5695 . . . . . . 7  |-  ( F : X -1-1-onto-> Y  ->  ( `' F  o.  F )  =  (  _I  |`  X ) )
4342ad2antrl 709 . . . . . 6  |-  ( (
ph  /\  ( F : X -1-1-onto-> Y  /\  G  =  `' F ) )  -> 
( `' F  o.  F )  =  (  _I  |`  X )
)
4441, 43eqtrd 2467 . . . . 5  |-  ( (
ph  /\  ( F : X -1-1-onto-> Y  /\  G  =  `' F ) )  -> 
( G  o.  F
)  =  (  _I  |`  X ) )
4540coeq2d 5026 . . . . . 6  |-  ( (
ph  /\  ( F : X -1-1-onto-> Y  /\  G  =  `' F ) )  -> 
( F  o.  G
)  =  ( F  o.  `' F ) )
46 f1ococnv2 5693 . . . . . . 7  |-  ( F : X -1-1-onto-> Y  ->  ( F  o.  `' F )  =  (  _I  |`  Y )
)
4746ad2antrl 709 . . . . . 6  |-  ( (
ph  /\  ( F : X -1-1-onto-> Y  /\  G  =  `' F ) )  -> 
( F  o.  `' F )  =  (  _I  |`  Y )
)
4845, 47eqtrd 2467 . . . . 5  |-  ( (
ph  /\  ( F : X -1-1-onto-> Y  /\  G  =  `' F ) )  -> 
( F  o.  G
)  =  (  _I  |`  Y ) )
4944, 48jca 519 . . . 4  |-  ( (
ph  /\  ( F : X -1-1-onto-> Y  /\  G  =  `' F ) )  -> 
( ( G  o.  F )  =  (  _I  |`  X )  /\  ( F  o.  G
)  =  (  _I  |`  Y ) ) )
5032, 39, 49jca31 521 . . 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 806 . 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 271 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 177    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725   class class class wbr 4204    _I cid 4485   `'ccnv 4868    |` cres 4871    o. ccom 4873   -->wf 5441   -1-1-onto->wf1o 5444   ` cfv 5445  (class class class)co 6072   Basecbs 13457   Catccat 13877  Sectcsect 13958  Invcinv 13959   SetCatcsetc 14218
This theorem is referenced by:  setciso  14234  yonedainv  14366
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4692  ax-cnex 9035  ax-resscn 9036  ax-1cn 9037  ax-icn 9038  ax-addcl 9039  ax-addrcl 9040  ax-mulcl 9041  ax-mulrcl 9042  ax-mulcom 9043  ax-addass 9044  ax-mulass 9045  ax-distr 9046  ax-i2m1 9047  ax-1ne0 9048  ax-1rid 9049  ax-rnegex 9050  ax-rrecex 9051  ax-cnre 9052  ax-pre-lttri 9053  ax-pre-lttrn 9054  ax-pre-ltadd 9055  ax-pre-mulgt0 9056
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-int 4043  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4837  df-xp 4875  df-rel 4876  df-cnv 4877  df-co 4878  df-dm 4879  df-rn 4880  df-res 4881  df-ima 4882  df-iota 5409  df-fun 5447  df-fn 5448  df-f 5449  df-f1 5450  df-fo 5451  df-f1o 5452  df-fv 5453  df-ov 6075  df-oprab 6076  df-mpt2 6077  df-1st 6340  df-2nd 6341  df-riota 6540  df-recs 6624  df-rdg 6659  df-1o 6715  df-oadd 6719  df-er 6896  df-map 7011  df-en 7101  df-dom 7102  df-sdom 7103  df-fin 7104  df-pnf 9111  df-mnf 9112  df-xr 9113  df-ltxr 9114  df-le 9115  df-sub 9282  df-neg 9283  df-nn 9990  df-2 10047  df-3 10048  df-4 10049  df-5 10050  df-6 10051  df-7 10052  df-8 10053  df-9 10054  df-10 10055  df-n0 10211  df-z 10272  df-dec 10372  df-uz 10478  df-fz 11033  df-struct 13459  df-ndx 13460  df-slot 13461  df-base 13462  df-hom 13541  df-cco 13542  df-cat 13881  df-cid 13882  df-sect 13961  df-inv 13962  df-setc 14219
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