Theorem issect 14998

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

Dummy variables f g are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		issect.b	. . . 4 |- B = ( Base ` C )
2		issect.h	. . . 4 |- H = ( Hom ` C )
3		issect.o	. . . 4 |- .x. = (comp ` C )
4		issect.i	. . . 4 |- .1. = ( Id ` C )
5		issect.s	. . . 4 |- S = (Sect ` C )
6		issect.c	. . . 4 |- ( ph -> C e. Cat )
7		issect.x	. . . 4 |- ( ph -> X e. B )
8		issect.y	. . . 4 |- ( ph -> Y e. B )
9	1, 2, 3, 4, 5, 6, 7, 8	sectfval 14996	. . 3 |- ( 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 4451	. 2 |- ( 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 6284	. . . . . 6 |- ( ( g = G /\ f = F ) -> ( g ( <. X , Y >. .x. X ) f ) = ( G ( <. X , Y >. .x. X ) F ) )
12	11	ancoms 453	. . . . 5 |- ( ( f = F /\ g = G ) -> ( g ( <. X , Y >. .x. X ) f ) = ( G ( <. X , Y >. .x. X ) F ) )
13	12	eqeq1d 2462	. . . 4 |- ( ( f = F /\ g = G ) -> ( ( g ( <. X , Y >. .x. X ) f ) = ( .1. ` X ) <-> ( G ( <. X , Y >. .x. X ) F ) = ( .1. ` X ) ) )
14		eqid 2460	. . . 4 |- { <. 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 5041	. . 3 |- ( 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 970	. . 3 |- ( ( 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 252	. 2 |- ( 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	syl6bb 261	1 |- ( ph -> ( F ( X S Y ) G <-> ( F e. ( X H Y ) /\ G e. ( Y H X ) /\ ( G ( <. X , Y >. .x. X ) F ) = ( .1. ` X ) ) ) )

Syntax hints:   -> wi 4   <-> wb 184   /\ wa 369   /\ w3a 968   = wceq 1374   e. wcel 1762   <.cop 4026   class class class wbr 4440   {copab 4497   ` cfv 5579  (class class class)co 6275   Basecbs 14479   Hom chom 14555  compcco 14556   Catccat 14908   Idccid 14909  Sectcsect 14989

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 1961  ax-ext 2438  ax-rep 4551  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567

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 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-op 4027  df-uni 4239  df-iun 4320  df-br 4441  df-opab 4499  df-mpt 4500  df-id 4788  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-ov 6278  df-oprab 6279  df-mpt2 6280  df-1st 6774  df-2nd 6775  df-sect 14992

This theorem is referenced by:  issect2  14999  sectcan  15000  sectco  15001  oppcsect  15018  sectmon  15022  monsect  15023  funcsect  15088  fucsect  15188  invfuc  15190  setcsect  15263  catciso  15281