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Theorem sectcan 13658
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 2283 . . . 4  |-  (  Hom  `  C )  =  (  Hom  `  C )
3 eqid 2283 . . . 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 2283 . . . . . . 7  |-  ( Id
`  C )  =  ( Id `  C
)
9 sectcan.s . . . . . . 7  |-  S  =  (Sect `  C )
101, 2, 3, 8, 9, 4, 5, 6issect 13656 . . . . . 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 201 . . . . 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 967 . . . 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 13656 . . . . . 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 201 . . . . 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 967 . . . 4  |-  ( ph  ->  F  e.  ( Y (  Hom  `  C
) X ) )
1715simp2d 968 . . . 4  |-  ( ph  ->  H  e.  ( X (  Hom  `  C
) Y ) )
181, 2, 3, 4, 5, 6, 5, 12, 16, 6, 17catass 13588 . . 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 969 . . . 4  |-  ( ph  ->  ( H ( <. Y ,  X >. (comp `  C ) Y ) F )  =  ( ( Id `  C
) `  Y )
)
2019oveq1d 5873 . . 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 969 . . . 4  |-  ( ph  ->  ( F ( <. X ,  Y >. (comp `  C ) X ) G )  =  ( ( Id `  C
) `  X )
)
2221oveq2d 5874 . . 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 2323 . 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 13585 . 2  |-  ( ph  ->  ( ( ( Id
`  C ) `  Y ) ( <. X ,  Y >. (comp `  C ) Y ) G )  =  G )
251, 2, 8, 4, 5, 3, 6, 17catrid 13586 . 2  |-  ( ph  ->  ( H ( <. X ,  X >. (comp `  C ) Y ) ( ( Id `  C ) `  X
) )  =  H )
2623, 24, 253eqtr3d 2323 1  |-  ( ph  ->  G  =  H )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ w3a 934    = wceq 1623    e. wcel 1684   <.cop 3643   class class class wbr 4023   ` cfv 5255  (class class class)co 5858   Basecbs 13148    Hom chom 13219  compcco 13220   Catccat 13566   Idccid 13567  Sectcsect 13647
This theorem is referenced by:  invfun  13666
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
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  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-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-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  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-cat 13570  df-cid 13571  df-sect 13650
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