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Theorem sectss 14999
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 14998 . 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 5067 . 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 3549 1  |-  ( ph  ->  ( X S Y )  C_  ( ( X H Y )  X.  ( Y H X ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1374    e. wcel 1762    C_ wss 3471   <.cop 4028   {copab 4499    X. cxp 4992   ` cfv 5581  (class class class)co 6277   Basecbs 14481   Hom chom 14557  compcco 14558   Catccat 14910   Idccid 14911  Sectcsect 14991
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1963  ax-ext 2440  ax-rep 4553  ax-sep 4563  ax-nul 4571  ax-pow 4620  ax-pr 4681  ax-un 6569
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2274  df-mo 2275  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2612  df-ne 2659  df-ral 2814  df-rex 2815  df-reu 2816  df-rab 2818  df-v 3110  df-sbc 3327  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3781  df-if 3935  df-pw 4007  df-sn 4023  df-pr 4025  df-op 4029  df-uni 4241  df-iun 4322  df-br 4443  df-opab 4501  df-mpt 4502  df-id 4790  df-xp 5000  df-rel 5001  df-cnv 5002  df-co 5003  df-dm 5004  df-rn 5005  df-res 5006  df-ima 5007  df-iota 5544  df-fun 5583  df-fn 5584  df-f 5585  df-f1 5586  df-fo 5587  df-f1o 5588  df-fv 5589  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-1st 6776  df-2nd 6777  df-sect 14994
This theorem is referenced by:  isinv  15006  invss  15007  oppcsect2  15021  oppcinv  15022
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