Theorem issect2

Description: Property of being a section. (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 )
	issect.f	|- ( ph -> F e. ( X H Y ) )
	issect.g	|- ( ph -> G e. ( Y H X ) )
Assertion	issect2	|- ( ph -> ( F ( X S Y ) G <-> ( G ( <. X , Y >. .x. X ) F ) = ( .1. ` 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		issect.f	|- ( ph -> F e. ( X H Y ) )
10		issect.g	|- ( ph -> G e. ( Y H X ) )
11	9 10	jca	|- ( ph -> ( F e. ( X H Y ) /\ G e. ( Y H X ) ) )
12	1 2 3 4 5 6 7 8	issect	|- ( ph -> ( F ( X S Y ) G <-> ( F e. ( X H Y ) /\ G e. ( Y H X ) /\ ( G ( <. X , Y >. .x. X ) F ) = ( .1. ` X ) ) ) )
13		df-3an	|- ( ( F e. ( X H Y ) /\ G e. ( Y H X ) /\ ( G ( <. X , Y >. .x. X ) F ) = ( .1. ` X ) ) <-> ( ( F e. ( X H Y ) /\ G e. ( Y H X ) ) /\ ( G ( <. X , Y >. .x. X ) F ) = ( .1. ` X ) ) )
14	12 13	bitrdi	|- ( ph -> ( F ( X S Y ) G <-> ( ( F e. ( X H Y ) /\ G e. ( Y H X ) ) /\ ( G ( <. X , Y >. .x. X ) F ) = ( .1. ` X ) ) ) )
15	11 14	mpbirand	|- ( ph -> ( F ( X S Y ) G <-> ( G ( <. X , Y >. .x. X ) F ) = ( .1. ` X ) ) )