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Metamath Proof Explorer

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 ) ) )