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Theorem sectcan 15171
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 2457 . . . 4  |-  ( Hom  `  C )  =  ( Hom  `  C )
3 eqid 2457 . . . 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 2457 . . . . . . 7  |-  ( Id
`  C )  =  ( Id `  C
)
9 sectcan.s . . . . . . 7  |-  S  =  (Sect `  C )
101, 2, 3, 8, 9, 4, 5, 6issect 15169 . . . . . 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 210 . . . . 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 1008 . . . 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 15169 . . . . . 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 210 . . . . 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 1008 . . . 4  |-  ( ph  ->  F  e.  ( Y ( Hom  `  C
) X ) )
1715simp2d 1009 . . . 4  |-  ( ph  ->  H  e.  ( X ( Hom  `  C
) Y ) )
181, 2, 3, 4, 5, 6, 5, 12, 16, 6, 17catass 15103 . . 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 1010 . . . 4  |-  ( ph  ->  ( H ( <. Y ,  X >. (comp `  C ) Y ) F )  =  ( ( Id `  C
) `  Y )
)
2019oveq1d 6311 . . 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 1010 . . . 4  |-  ( ph  ->  ( F ( <. X ,  Y >. (comp `  C ) X ) G )  =  ( ( Id `  C
) `  X )
)
2221oveq2d 6312 . . 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 2506 . 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 15100 . 2  |-  ( ph  ->  ( ( ( Id
`  C ) `  Y ) ( <. X ,  Y >. (comp `  C ) Y ) G )  =  G )
251, 2, 8, 4, 5, 3, 6, 17catrid 15101 . 2  |-  ( ph  ->  ( H ( <. X ,  X >. (comp `  C ) Y ) ( ( Id `  C ) `  X
) )  =  H )
2623, 24, 253eqtr3d 2506 1  |-  ( ph  ->  G  =  H )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ w3a 973    = wceq 1395    e. wcel 1819   <.cop 4038   class class class wbr 4456   ` cfv 5594  (class class class)co 6296   Basecbs 14644   Hom chom 14723  compcco 14724   Catccat 15081   Idccid 15082  Sectcsect 15160
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-id 4804  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-1st 6799  df-2nd 6800  df-cat 15085  df-cid 15086  df-sect 15163
This theorem is referenced by:  invfun  15180  inveq  15190
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