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Theorem setcsect 13937
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 2296 . . 3  |-  ( Base `  C )  =  (
Base `  C )
2 eqid 2296 . . 3  |-  (  Hom  `  C )  =  (  Hom  `  C )
3 eqid 2296 . . 3  |-  (comp `  C )  =  (comp `  C )
4 eqid 2296 . . 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 13933 . . . 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 13926 . . . 4  |-  ( ph  ->  U  =  ( Base `  C ) )
1210, 11eleqtrd 2372 . . 3  |-  ( ph  ->  X  e.  ( Base `  C ) )
13 setcmon.y . . . 4  |-  ( ph  ->  Y  e.  U )
1413, 11eleqtrd 2372 . . 3  |-  ( ph  ->  Y  e.  ( Base `  C ) )
151, 2, 3, 4, 5, 9, 12, 14issect 13672 . 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 13929 . . . . . 6  |-  ( ph  ->  ( F  e.  ( X (  Hom  `  C
) Y )  <->  F : X
--> Y ) )
177, 6, 2, 13, 10elsetchom 13929 . . . . . 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 13931 . . . . . 6  |-  ( (
ph  /\  ( F : X --> Y  /\  G : Y --> X ) )  ->  ( G (
<. X ,  Y >. (comp `  C ) X ) F )  =  ( G  o.  F ) )
267, 4, 6, 10setcid 13934 . . . . . . 7  |-  ( ph  ->  ( ( Id `  C ) `  X
)  =  (  _I  |`  X ) )
2726adantr 451 . . . . . 6  |-  ( (
ph  /\  ( F : X --> Y  /\  G : Y --> X ) )  ->  ( ( Id
`  C ) `  X )  =  (  _I  |`  X )
)
2825, 27eqeq12d 2310 . . . . 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 1632    e. wcel 1696   <.cop 3656   class class class wbr 4039    _I cid 4320    |` cres 4707    o. ccom 4709   -->wf 5267   ` cfv 5271  (class class class)co 5874   Basecbs 13164    Hom chom 13235  compcco 13236   Catccat 13582   Idccid 13583  Sectcsect 13663   SetCatcsetc 13923
This theorem is referenced by:  setcinv  13938
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-1o 6495  df-oadd 6499  df-er 6676  df-map 6790  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-nn 9763  df-2 9820  df-3 9821  df-4 9822  df-5 9823  df-6 9824  df-7 9825  df-8 9826  df-9 9827  df-10 9828  df-n0 9982  df-z 10041  df-dec 10141  df-uz 10247  df-fz 10799  df-struct 13166  df-ndx 13167  df-slot 13168  df-base 13169  df-hom 13248  df-cco 13249  df-cat 13586  df-cid 13587  df-sect 13666  df-setc 13924
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