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Theorem setcsect 13921
Description: A section in the category of sets, written out. (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 )
setcsect.n  |-  S  =  (Sect `  C )
Assertion
Ref Expression
setcsect  |-  ( ph  ->  ( F ( X S Y ) G  <-> 
( F : X --> Y  /\  G : Y --> X  /\  ( G  o.  F )  =  (  _I  |`  X )
) ) )

Proof of Theorem setcsect
StepHypRef Expression
1 eqid 2283 . . 3  |-  ( Base `  C )  =  (
Base `  C )
2 eqid 2283 . . 3  |-  (  Hom  `  C )  =  (  Hom  `  C )
3 eqid 2283 . . 3  |-  (comp `  C )  =  (comp `  C )
4 eqid 2283 . . 3  |-  ( Id
`  C )  =  ( Id `  C
)
5 setcsect.n . . 3  |-  S  =  (Sect `  C )
6 setcmon.u . . . 4  |-  ( ph  ->  U  e.  V )
7 setcmon.c . . . . 5  |-  C  =  ( SetCat `  U )
87setccat 13917 . . . 4  |-  ( U  e.  V  ->  C  e.  Cat )
96, 8syl 15 . . 3  |-  ( ph  ->  C  e.  Cat )
10 setcmon.x . . . 4  |-  ( ph  ->  X  e.  U )
117, 6setcbas 13910 . . . 4  |-  ( ph  ->  U  =  ( Base `  C ) )
1210, 11eleqtrd 2359 . . 3  |-  ( ph  ->  X  e.  ( Base `  C ) )
13 setcmon.y . . . 4  |-  ( ph  ->  Y  e.  U )
1413, 11eleqtrd 2359 . . 3  |-  ( ph  ->  Y  e.  ( Base `  C ) )
151, 2, 3, 4, 5, 9, 12, 14issect 13656 . 2  |-  ( ph  ->  ( F ( X S Y ) G  <-> 
( F  e.  ( X (  Hom  `  C
) Y )  /\  G  e.  ( Y
(  Hom  `  C ) X )  /\  ( G ( <. X ,  Y >. (comp `  C
) X ) F )  =  ( ( Id `  C ) `
 X ) ) ) )
167, 6, 2, 10, 13elsetchom 13913 . . . . . 6  |-  ( ph  ->  ( F  e.  ( X (  Hom  `  C
) Y )  <->  F : X
--> Y ) )
177, 6, 2, 13, 10elsetchom 13913 . . . . . 6  |-  ( ph  ->  ( G  e.  ( Y (  Hom  `  C
) X )  <->  G : Y
--> X ) )
1816, 17anbi12d 691 . . . . 5  |-  ( ph  ->  ( ( F  e.  ( X (  Hom  `  C ) Y )  /\  G  e.  ( Y (  Hom  `  C
) X ) )  <-> 
( F : X --> Y  /\  G : Y --> X ) ) )
1918anbi1d 685 . . . 4  |-  ( ph  ->  ( ( ( F  e.  ( X (  Hom  `  C ) Y )  /\  G  e.  ( Y (  Hom  `  C ) X ) )  /\  ( G ( <. X ,  Y >. (comp `  C ) X ) F )  =  ( ( Id
`  C ) `  X ) )  <->  ( ( F : X --> Y  /\  G : Y --> X )  /\  ( G (
<. X ,  Y >. (comp `  C ) X ) F )  =  ( ( Id `  C
) `  X )
) ) )
206adantr 451 . . . . . . 7  |-  ( (
ph  /\  ( F : X --> Y  /\  G : Y --> X ) )  ->  U  e.  V
)
2110adantr 451 . . . . . . 7  |-  ( (
ph  /\  ( F : X --> Y  /\  G : Y --> X ) )  ->  X  e.  U
)
2213adantr 451 . . . . . . 7  |-  ( (
ph  /\  ( F : X --> Y  /\  G : Y --> X ) )  ->  Y  e.  U
)
23 simprl 732 . . . . . . 7  |-  ( (
ph  /\  ( F : X --> Y  /\  G : Y --> X ) )  ->  F : X --> Y )
24 simprr 733 . . . . . . 7  |-  ( (
ph  /\  ( F : X --> Y  /\  G : Y --> X ) )  ->  G : Y --> X )
257, 20, 3, 21, 22, 21, 23, 24setcco 13915 . . . . . 6  |-  ( (
ph  /\  ( F : X --> Y  /\  G : Y --> X ) )  ->  ( G (
<. X ,  Y >. (comp `  C ) X ) F )  =  ( G  o.  F ) )
267, 4, 6, 10setcid 13918 . . . . . . 7  |-  ( ph  ->  ( ( Id `  C ) `  X
)  =  (  _I  |`  X ) )
2726adantr 451 . . . . . 6  |-  ( (
ph  /\  ( F : X --> Y  /\  G : Y --> X ) )  ->  ( ( Id
`  C ) `  X )  =  (  _I  |`  X )
)
2825, 27eqeq12d 2297 . . . . 5  |-  ( (
ph  /\  ( F : X --> Y  /\  G : Y --> X ) )  ->  ( ( G ( <. X ,  Y >. (comp `  C ) X ) F )  =  ( ( Id
`  C ) `  X )  <->  ( G  o.  F )  =  (  _I  |`  X )
) )
2928pm5.32da 622 . . . 4  |-  ( ph  ->  ( ( ( F : X --> Y  /\  G : Y --> X )  /\  ( G (
<. X ,  Y >. (comp `  C ) X ) F )  =  ( ( Id `  C
) `  X )
)  <->  ( ( F : X --> Y  /\  G : Y --> X )  /\  ( G  o.  F )  =  (  _I  |`  X )
) ) )
3019, 29bitrd 244 . . 3  |-  ( ph  ->  ( ( ( F  e.  ( X (  Hom  `  C ) Y )  /\  G  e.  ( Y (  Hom  `  C ) X ) )  /\  ( G ( <. X ,  Y >. (comp `  C ) X ) F )  =  ( ( Id
`  C ) `  X ) )  <->  ( ( F : X --> Y  /\  G : Y --> X )  /\  ( G  o.  F )  =  (  _I  |`  X )
) ) )
31 df-3an 936 . . 3  |-  ( ( F  e.  ( X (  Hom  `  C
) Y )  /\  G  e.  ( Y
(  Hom  `  C ) X )  /\  ( G ( <. X ,  Y >. (comp `  C
) X ) F )  =  ( ( Id `  C ) `
 X ) )  <-> 
( ( F  e.  ( X (  Hom  `  C ) Y )  /\  G  e.  ( Y (  Hom  `  C
) X ) )  /\  ( G (
<. X ,  Y >. (comp `  C ) X ) F )  =  ( ( Id `  C
) `  X )
) )
32 df-3an 936 . . 3  |-  ( ( F : X --> Y  /\  G : Y --> X  /\  ( G  o.  F
)  =  (  _I  |`  X ) )  <->  ( ( F : X --> Y  /\  G : Y --> X )  /\  ( G  o.  F )  =  (  _I  |`  X )
) )
3330, 31, 323bitr4g 279 . 2  |-  ( ph  ->  ( ( F  e.  ( X (  Hom  `  C ) Y )  /\  G  e.  ( Y (  Hom  `  C
) X )  /\  ( G ( <. X ,  Y >. (comp `  C
) X ) F )  =  ( ( Id `  C ) `
 X ) )  <-> 
( F : X --> Y  /\  G : Y --> X  /\  ( G  o.  F )  =  (  _I  |`  X )
) ) )
3415, 33bitrd 244 1  |-  ( ph  ->  ( F ( X S Y ) G  <-> 
( F : X --> Y  /\  G : Y --> X  /\  ( G  o.  F )  =  (  _I  |`  X )
) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   <.cop 3643   class class class wbr 4023    _I cid 4304    |` cres 4691    o. ccom 4693   -->wf 5251   ` cfv 5255  (class class class)co 5858   Basecbs 13148    Hom chom 13219  compcco 13220   Catccat 13566   Idccid 13567  Sectcsect 13647   SetCatcsetc 13907
This theorem is referenced by:  setcinv  13922
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814
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 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-oadd 6483  df-er 6660  df-map 6774  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-nn 9747  df-2 9804  df-3 9805  df-4 9806  df-5 9807  df-6 9808  df-7 9809  df-8 9810  df-9 9811  df-10 9812  df-n0 9966  df-z 10025  df-dec 10125  df-uz 10231  df-fz 10783  df-struct 13150  df-ndx 13151  df-slot 13152  df-base 13153  df-hom 13232  df-cco 13233  df-cat 13570  df-cid 13571  df-sect 13650  df-setc 13908
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