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Theorem sectcan 13502
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 2253 . . . 4  |-  (  Hom  `  C )  =  (  Hom  `  C )
3 eqid 2253 . . . 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 2253 . . . . . . 7  |-  ( Id
`  C )  =  ( Id `  C
)
9 sectcan.s . . . . . . 7  |-  S  =  (Sect `  C )
101, 2, 3, 8, 9, 4, 5, 6issect 13500 . . . . . 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 972 . . . 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 13500 . . . . . 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 972 . . . 4  |-  ( ph  ->  F  e.  ( Y (  Hom  `  C
) X ) )
1715simp2d 973 . . . 4  |-  ( ph  ->  H  e.  ( X (  Hom  `  C
) Y ) )
181, 2, 3, 4, 5, 6, 5, 12, 16, 6, 17catass 13432 . . 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 974 . . . 4  |-  ( ph  ->  ( H ( <. Y ,  X >. (comp `  C ) Y ) F )  =  ( ( Id `  C
) `  Y )
)
2019oveq1d 5725 . . 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 974 . . . 4  |-  ( ph  ->  ( F ( <. X ,  Y >. (comp `  C ) X ) G )  =  ( ( Id `  C
) `  X )
)
2221oveq2d 5726 . . 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 2293 . 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 13429 . 2  |-  ( ph  ->  ( ( ( Id
`  C ) `  Y ) ( <. X ,  Y >. (comp `  C ) Y ) G )  =  G )
251, 2, 8, 4, 5, 3, 6, 17catrid 13430 . 2  |-  ( ph  ->  ( H ( <. X ,  X >. (comp `  C ) Y ) ( ( Id `  C ) `  X
) )  =  H )
2623, 24, 253eqtr3d 2293 1  |-  ( ph  ->  G  =  H )
Colors of variables: wff set class
Syntax hints:    -> wi 6    /\ w3a 939    = wceq 1619    e. wcel 1621   <.cop 3547   class class class wbr 3920   ` cfv 4592  (class class class)co 5710   Basecbs 13022    Hom chom 13093  compcco 13094   Catccat 13410   Idccid 13411  Sectcsect 13491
This theorem is referenced by:  invfun  13510
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1926  ax-ext 2234  ax-rep 4028  ax-sep 4038  ax-nul 4046  ax-pow 4082  ax-pr 4108  ax-un 4403
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1883  df-eu 2118  df-mo 2119  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-ne 2414  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2516  df-v 2729  df-sbc 2922  df-csb 3010  df-dif 3081  df-un 3083  df-in 3085  df-ss 3089  df-nul 3363  df-if 3471  df-pw 3532  df-sn 3550  df-pr 3551  df-op 3553  df-uni 3728  df-iun 3805  df-br 3921  df-opab 3975  df-mpt 3976  df-id 4202  df-xp 4594  df-rel 4595  df-cnv 4596  df-co 4597  df-dm 4598  df-rn 4599  df-res 4600  df-ima 4601  df-fun 4602  df-fn 4603  df-f 4604  df-f1 4605  df-fo 4606  df-f1o 4607  df-fv 4608  df-ov 5713  df-oprab 5714  df-mpt2 5715  df-1st 5974  df-2nd 5975  df-iota 6143  df-riota 6190  df-cat 13414  df-cid 13415  df-sect 13494
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