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Theorem sectcan 13652
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 2284 . . . 4  |-  (  Hom  `  C )  =  (  Hom  `  C )
3 eqid 2284 . . . 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 2284 . . . . . . 7  |-  ( Id
`  C )  =  ( Id `  C
)
9 sectcan.s . . . . . . 7  |-  S  =  (Sect `  C )
101, 2, 3, 8, 9, 4, 5, 6issect 13650 . . . . . 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 203 . . . . 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 13650 . . . . . 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 203 . . . . 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 13582 . . 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 5834 . . 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 5835 . . 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 2324 . 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 13579 . 2  |-  ( ph  ->  ( ( ( Id
`  C ) `  Y ) ( <. X ,  Y >. (comp `  C ) Y ) G )  =  G )
251, 2, 8, 4, 5, 3, 6, 17catrid 13580 . 2  |-  ( ph  ->  ( H ( <. X ,  X >. (comp `  C ) Y ) ( ( Id `  C ) `  X
) )  =  H )
2623, 24, 253eqtr3d 2324 1  |-  ( ph  ->  G  =  H )
Colors of variables: wff set class
Syntax hints:    -> wi 6    /\ w3a 936    = wceq 1624    e. wcel 1685   <.cop 3644   class class class wbr 4024   ` cfv 5221  (class class class)co 5819   Basecbs 13142    Hom chom 13213  compcco 13214   Catccat 13560   Idccid 13561  Sectcsect 13641
This theorem is referenced by:  invfun  13660
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1534  ax-5 1545  ax-17 1604  ax-9 1637  ax-8 1645  ax-13 1687  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1867  ax-ext 2265  ax-rep 4132  ax-sep 4142  ax-nul 4150  ax-pow 4187  ax-pr 4213  ax-un 4511
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 938  df-tru 1312  df-ex 1530  df-nf 1533  df-sb 1632  df-eu 2148  df-mo 2149  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-ral 2549  df-rex 2550  df-reu 2551  df-rmo 2552  df-rab 2553  df-v 2791  df-sbc 2993  df-csb 3083  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3457  df-if 3567  df-pw 3628  df-sn 3647  df-pr 3648  df-op 3650  df-uni 3829  df-iun 3908  df-br 4025  df-opab 4079  df-mpt 4080  df-id 4308  df-xp 4694  df-rel 4695  df-cnv 4696  df-co 4697  df-dm 4698  df-rn 4699  df-res 4700  df-ima 4701  df-fun 5223  df-fn 5224  df-f 5225  df-f1 5226  df-fo 5227  df-f1o 5228  df-fv 5229  df-ov 5822  df-oprab 5823  df-mpt2 5824  df-1st 6083  df-2nd 6084  df-iota 6252  df-riota 6299  df-cat 13564  df-cid 13565  df-sect 13644
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