Metamath Proof Explorer

Theorem issect

Description: The property " F is a section of G ". (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 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 ) ) ) )

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 9 breqd 
 |-  ( ph -> ( F ( X S Y ) G <-> F { <. f , g >. | ( ( f e. ( X H Y ) /\ g e. ( Y H X ) ) /\ ( g ( <. X , Y >. .x. X ) f ) = ( .1. ` X ) ) } G ) )
11 oveq12 
 |-  ( ( g = G /\ f = F ) -> ( g ( <. X , Y >. .x. X ) f ) = ( G ( <. X , Y >. .x. X ) F ) )
12 11 ancoms 
 |-  ( ( f = F /\ g = G ) -> ( g ( <. X , Y >. .x. X ) f ) = ( G ( <. X , Y >. .x. X ) F ) )
13 12 eqeq1d 
 |-  ( ( f = F /\ g = G ) -> ( ( g ( <. X , Y >. .x. X ) f ) = ( .1. ` X ) <-> ( G ( <. X , Y >. .x. X ) F ) = ( .1. ` X ) ) )
14 eqid 
 |-  { <. f , g >. | ( ( f e. ( X H Y ) /\ g e. ( Y H X ) ) /\ ( g ( <. X , Y >. .x. X ) f ) = ( .1. ` X ) ) } = { <. f , g >. | ( ( f e. ( X H Y ) /\ g e. ( Y H X ) ) /\ ( g ( <. X , Y >. .x. X ) f ) = ( .1. ` X ) ) }
15 13 14 brab2a 
 |-  ( F { <. f , g >. | ( ( f e. ( X H Y ) /\ g e. ( Y H X ) ) /\ ( g ( <. X , Y >. .x. X ) f ) = ( .1. ` X ) ) } G <-> ( ( F e. ( X H Y ) /\ G e. ( Y H X ) ) /\ ( G ( <. X , Y >. .x. X ) F ) = ( .1. ` X ) ) )
16 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 ) ) )
17 15 16 bitr4i 
 |-  ( F { <. f , g >. | ( ( f e. ( X H Y ) /\ g e. ( Y H X ) ) /\ ( g ( <. X , Y >. .x. X ) f ) = ( .1. ` X ) ) } G <-> ( F e. ( X H Y ) /\ G e. ( Y H X ) /\ ( G ( <. X , Y >. .x. X ) F ) = ( .1. ` X ) ) )
18 10 17 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 ) ) ) )