Theorem issect 14711

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 14709	. . 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 4322	. 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 6119	. . . . . 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 2451	. . . 4 |- ( ( f = F /\ g = G ) -> ( ( g ( <. X , Y >. .x. X ) f ) = ( .1. ` X ) <-> ( G ( <. X , Y >. .x. X ) F ) = ( .1. ` X ) ) )
14		eqid 2443	. . . 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 4907	. . 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 967	. . 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 965   = wceq 1369   e. wcel 1756   <.cop 3902   class class class wbr 4311   {copab 4368   ` cfv 5437  (class class class)co 6110   Basecbs 14193   Hom chom 14268  compcco 14269   Catccat 14621   Idccid 14622  Sectcsect 14702

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4422  ax-sep 4432  ax-nul 4440  ax-pow 4489  ax-pr 4550  ax-un 6391

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2622  df-ral 2739  df-rex 2740  df-reu 2741  df-rab 2743  df-v 2993  df-sbc 3206  df-csb 3308  df-dif 3350  df-un 3352  df-in 3354  df-ss 3361  df-nul 3657  df-if 3811  df-pw 3881  df-sn 3897  df-pr 3899  df-op 3903  df-uni 4111  df-iun 4192  df-br 4312  df-opab 4370  df-mpt 4371  df-id 4655  df-xp 4865  df-rel 4866  df-cnv 4867  df-co 4868  df-dm 4869  df-rn 4870  df-res 4871  df-ima 4872  df-iota 5400  df-fun 5439  df-fn 5440  df-f 5441  df-f1 5442  df-fo 5443  df-f1o 5444  df-fv 5445  df-ov 6113  df-oprab 6114  df-mpt2 6115  df-1st 6596  df-2nd 6597  df-sect 14705

This theorem is referenced by:  issect2  14712  sectcan  14713  sectco  14714  oppcsect  14731  sectmon  14735  monsect  14736  funcsect  14801  fucsect  14901  invfuc  14903  setcsect  14976  catciso  14994