Theorem issect 15658

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 15656	. . 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 4413	. 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 6299	. . . . . 6 |- ( ( g = G /\ f = F ) -> ( g ( <. X , Y >. .x. X ) f ) = ( G ( <. X , Y >. .x. X ) F ) )
12	11	ancoms 455	. . . . 5 |- ( ( f = F /\ g = G ) -> ( g ( <. X , Y >. .x. X ) f ) = ( G ( <. X , Y >. .x. X ) F ) )
13	12	eqeq1d 2453	. . . 4 |- ( ( f = F /\ g = G ) -> ( ( g ( <. X , Y >. .x. X ) f ) = ( .1. ` X ) <-> ( G ( <. X , Y >. .x. X ) F ) = ( .1. ` X ) ) )
14		eqid 2451	. . . 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 4884	. . 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 987	. . 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 256	. 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 265	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 188   /\ wa 371   /\ w3a 985   = wceq 1444   e. wcel 1887   <.cop 3974   class class class wbr 4402   {copab 4460   ` cfv 5582  (class class class)co 6290   Basecbs 15121   Hom chom 15201  compcco 15202   Catccat 15570   Idccid 15571  Sectcsect 15649

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-rep 4515  ax-sep 4525  ax-nul 4534  ax-pow 4581  ax-pr 4639  ax-un 6583

This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-ral 2742  df-rex 2743  df-reu 2744  df-rab 2746  df-v 3047  df-sbc 3268  df-csb 3364  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-nul 3732  df-if 3882  df-pw 3953  df-sn 3969  df-pr 3971  df-op 3975  df-uni 4199  df-iun 4280  df-br 4403  df-opab 4462  df-mpt 4463  df-id 4749  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-res 4846  df-ima 4847  df-iota 5546  df-fun 5584  df-fn 5585  df-f 5586  df-f1 5587  df-fo 5588  df-f1o 5589  df-fv 5590  df-ov 6293  df-oprab 6294  df-mpt2 6295  df-1st 6793  df-2nd 6794  df-sect 15652

This theorem is referenced by:  issect2  15659  sectcan  15660  sectco  15661  oppcsect  15683  sectmon  15687  monsect  15688  funcsect  15777  fucsect  15877  invfuc  15879  setcsect  15984  catciso  16002  rngcsect  40035  rngcsectALTV  40047  ringcsect  40086  ringcsectALTV  40110