MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  sectss Unicode version

Theorem sectss 13907
Description: The section relation is a relation between morphisms from  X to  Y and morphisms from  Y to  X. (Contributed by Mario Carneiro, 2-Jan-2017.)
Hypotheses
Ref Expression
issect.b  |-  B  =  ( Base `  C
)
issect.h  |-  H  =  (  Hom  `  C
)
issect.o  |-  .x.  =  (comp `  C )
issect.i  |-  .1.  =  ( Id `  C )
issect.s  |-  S  =  (Sect `  C )
issect.c  |-  ( ph  ->  C  e.  Cat )
issect.x  |-  ( ph  ->  X  e.  B )
issect.y  |-  ( ph  ->  Y  e.  B )
Assertion
Ref Expression
sectss  |-  ( ph  ->  ( X S Y )  C_  ( ( X H Y )  X.  ( Y H X ) ) )

Proof of Theorem sectss
Dummy variables  f 
g are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 issect.b . . 3  |-  B  =  ( Base `  C
)
2 issect.h . . 3  |-  H  =  (  Hom  `  C
)
3 issect.o . . 3  |-  .x.  =  (comp `  C )
4 issect.i . . 3  |-  .1.  =  ( Id `  C )
5 issect.s . . 3  |-  S  =  (Sect `  C )
6 issect.c . . 3  |-  ( ph  ->  C  e.  Cat )
7 issect.x . . 3  |-  ( ph  ->  X  e.  B )
8 issect.y . . 3  |-  ( ph  ->  Y  e.  B )
91, 2, 3, 4, 5, 6, 7, 8sectfval 13906 . 2  |-  ( ph  ->  ( X S Y )  =  { <. f ,  g >.  |  ( ( f  e.  ( X H Y )  /\  g  e.  ( Y H X ) )  /\  ( g ( <. X ,  Y >.  .x.  X ) f )  =  (  .1.  `  X ) ) } )
10 opabssxp 4892 . 2  |-  { <. f ,  g >.  |  ( ( f  e.  ( X H Y )  /\  g  e.  ( Y H X ) )  /\  ( g ( <. X ,  Y >.  .x.  X ) f )  =  (  .1.  `  X ) ) } 
C_  ( ( X H Y )  X.  ( Y H X ) )
119, 10syl6eqss 3343 1  |-  ( ph  ->  ( X S Y )  C_  ( ( X H Y )  X.  ( Y H X ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1717    C_ wss 3265   <.cop 3762   {copab 4208    X. cxp 4818   ` cfv 5396  (class class class)co 6022   Basecbs 13398    Hom chom 13469  compcco 13470   Catccat 13818   Idccid 13819  Sectcsect 13899
This theorem is referenced by:  isinv  13914  invss  13915  oppcsect2  13929  oppcinv  13930
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 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2370  ax-rep 4263  ax-sep 4273  ax-nul 4281  ax-pow 4320  ax-pr 4346  ax-un 4643
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 2244  df-mo 2245  df-clab 2376  df-cleq 2382  df-clel 2385  df-nfc 2514  df-ne 2554  df-ral 2656  df-rex 2657  df-reu 2658  df-rab 2660  df-v 2903  df-sbc 3107  df-csb 3197  df-dif 3268  df-un 3270  df-in 3272  df-ss 3279  df-nul 3574  df-if 3685  df-pw 3746  df-sn 3765  df-pr 3766  df-op 3768  df-uni 3960  df-iun 4039  df-br 4156  df-opab 4210  df-mpt 4211  df-id 4441  df-xp 4826  df-rel 4827  df-cnv 4828  df-co 4829  df-dm 4830  df-rn 4831  df-res 4832  df-ima 4833  df-iota 5360  df-fun 5398  df-fn 5399  df-f 5400  df-f1 5401  df-fo 5402  df-f1o 5403  df-fv 5404  df-ov 6025  df-oprab 6026  df-mpt2 6027  df-1st 6290  df-2nd 6291  df-sect 13902
  Copyright terms: Public domain W3C validator