Theorem sectss

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)

	Ref	Expression
Hypotheses	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	sectss	|- ( ph -> ( X S Y ) C_ ( ( X H Y ) X. ( Y H X ) ) )

Proof

Step	Hyp	Ref	Expression
1		issect.b	|- B = ( Base ` C )
2		issect.h	|- H = ( Hom ` C )
3		issect.o	|- .x. = ( comp ` C )
4		issect.i	|- .1. = ( Id ` C )
5		issect.s	|- S = ( Sect ` C )
6		issect.c	|- ( ph -> C e. Cat )
7		issect.x	|- ( ph -> X e. B )
8		issect.y	|- ( ph -> Y e. B )
9	1 2 3 4 5 6 7 8	sectfval	|- ( 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	|- { <. 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 ) )
11	9 10	eqsstrdi	|- ( ph -> ( X S Y ) C_ ( ( X H Y ) X. ( Y H X ) ) )