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Theorem sectcan 13936
Description: If  G is a section of  F and  F is a section of  H, then  G  =  H. Proposition 3.10 of [Adamek] p. 28. (Contributed by Mario Carneiro, 2-Jan-2017.)
Hypotheses
Ref Expression
sectcan.b  |-  B  =  ( Base `  C
)
sectcan.s  |-  S  =  (Sect `  C )
sectcan.c  |-  ( ph  ->  C  e.  Cat )
sectcan.x  |-  ( ph  ->  X  e.  B )
sectcan.y  |-  ( ph  ->  Y  e.  B )
sectcan.1  |-  ( ph  ->  G ( X S Y ) F )
sectcan.2  |-  ( ph  ->  F ( Y S X ) H )
Assertion
Ref Expression
sectcan  |-  ( ph  ->  G  =  H )

Proof of Theorem sectcan
StepHypRef Expression
1 sectcan.b . . . 4  |-  B  =  ( Base `  C
)
2 eqid 2404 . . . 4  |-  (  Hom  `  C )  =  (  Hom  `  C )
3 eqid 2404 . . . 4  |-  (comp `  C )  =  (comp `  C )
4 sectcan.c . . . 4  |-  ( ph  ->  C  e.  Cat )
5 sectcan.x . . . 4  |-  ( ph  ->  X  e.  B )
6 sectcan.y . . . 4  |-  ( ph  ->  Y  e.  B )
7 sectcan.1 . . . . . 6  |-  ( ph  ->  G ( X S Y ) F )
8 eqid 2404 . . . . . . 7  |-  ( Id
`  C )  =  ( Id `  C
)
9 sectcan.s . . . . . . 7  |-  S  =  (Sect `  C )
101, 2, 3, 8, 9, 4, 5, 6issect 13934 . . . . . 6  |-  ( ph  ->  ( G ( X S Y ) F  <-> 
( G  e.  ( X (  Hom  `  C
) Y )  /\  F  e.  ( Y
(  Hom  `  C ) X )  /\  ( F ( <. X ,  Y >. (comp `  C
) X ) G )  =  ( ( Id `  C ) `
 X ) ) ) )
117, 10mpbid 202 . . . . 5  |-  ( ph  ->  ( G  e.  ( X (  Hom  `  C
) Y )  /\  F  e.  ( Y
(  Hom  `  C ) X )  /\  ( F ( <. X ,  Y >. (comp `  C
) X ) G )  =  ( ( Id `  C ) `
 X ) ) )
1211simp1d 969 . . . 4  |-  ( ph  ->  G  e.  ( X (  Hom  `  C
) Y ) )
13 sectcan.2 . . . . . 6  |-  ( ph  ->  F ( Y S X ) H )
141, 2, 3, 8, 9, 4, 6, 5issect 13934 . . . . . 6  |-  ( ph  ->  ( F ( Y S X ) H  <-> 
( F  e.  ( Y (  Hom  `  C
) X )  /\  H  e.  ( X
(  Hom  `  C ) Y )  /\  ( H ( <. Y ,  X >. (comp `  C
) Y ) F )  =  ( ( Id `  C ) `
 Y ) ) ) )
1513, 14mpbid 202 . . . . 5  |-  ( ph  ->  ( F  e.  ( Y (  Hom  `  C
) X )  /\  H  e.  ( X
(  Hom  `  C ) Y )  /\  ( H ( <. Y ,  X >. (comp `  C
) Y ) F )  =  ( ( Id `  C ) `
 Y ) ) )
1615simp1d 969 . . . 4  |-  ( ph  ->  F  e.  ( Y (  Hom  `  C
) X ) )
1715simp2d 970 . . . 4  |-  ( ph  ->  H  e.  ( X (  Hom  `  C
) Y ) )
181, 2, 3, 4, 5, 6, 5, 12, 16, 6, 17catass 13866 . . 3  |-  ( ph  ->  ( ( H (
<. Y ,  X >. (comp `  C ) Y ) F ) ( <. X ,  Y >. (comp `  C ) Y ) G )  =  ( H ( <. X ,  X >. (comp `  C
) Y ) ( F ( <. X ,  Y >. (comp `  C
) X ) G ) ) )
1915simp3d 971 . . . 4  |-  ( ph  ->  ( H ( <. Y ,  X >. (comp `  C ) Y ) F )  =  ( ( Id `  C
) `  Y )
)
2019oveq1d 6055 . . 3  |-  ( ph  ->  ( ( H (
<. Y ,  X >. (comp `  C ) Y ) F ) ( <. X ,  Y >. (comp `  C ) Y ) G )  =  ( ( ( Id `  C ) `  Y
) ( <. X ,  Y >. (comp `  C
) Y ) G ) )
2111simp3d 971 . . . 4  |-  ( ph  ->  ( F ( <. X ,  Y >. (comp `  C ) X ) G )  =  ( ( Id `  C
) `  X )
)
2221oveq2d 6056 . . 3  |-  ( ph  ->  ( H ( <. X ,  X >. (comp `  C ) Y ) ( F ( <. X ,  Y >. (comp `  C ) X ) G ) )  =  ( H ( <. X ,  X >. (comp `  C ) Y ) ( ( Id `  C ) `  X
) ) )
2318, 20, 223eqtr3d 2444 . 2  |-  ( ph  ->  ( ( ( Id
`  C ) `  Y ) ( <. X ,  Y >. (comp `  C ) Y ) G )  =  ( H ( <. X ,  X >. (comp `  C
) Y ) ( ( Id `  C
) `  X )
) )
241, 2, 8, 4, 5, 3, 6, 12catlid 13863 . 2  |-  ( ph  ->  ( ( ( Id
`  C ) `  Y ) ( <. X ,  Y >. (comp `  C ) Y ) G )  =  G )
251, 2, 8, 4, 5, 3, 6, 17catrid 13864 . 2  |-  ( ph  ->  ( H ( <. X ,  X >. (comp `  C ) Y ) ( ( Id `  C ) `  X
) )  =  H )
2623, 24, 253eqtr3d 2444 1  |-  ( ph  ->  G  =  H )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ w3a 936    = wceq 1649    e. wcel 1721   <.cop 3777   class class class wbr 4172   ` cfv 5413  (class class class)co 6040   Basecbs 13424    Hom chom 13495  compcco 13496   Catccat 13844   Idccid 13845  Sectcsect 13925
This theorem is referenced by:  invfun  13944
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-id 4458  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-1st 6308  df-2nd 6309  df-riota 6508  df-cat 13848  df-cid 13849  df-sect 13928
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