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Theorem issect2 14705
Description: Property of being a section. (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 )
issect.f  |-  ( ph  ->  F  e.  ( X H Y ) )
issect.g  |-  ( ph  ->  G  e.  ( Y H X ) )
Assertion
Ref Expression
issect2  |-  ( ph  ->  ( F ( X S Y ) G  <-> 
( G ( <. X ,  Y >.  .x. 
X ) F )  =  (  .1.  `  X ) ) )

Proof of Theorem issect2
StepHypRef Expression
1 issect.f . . 3  |-  ( ph  ->  F  e.  ( X H Y ) )
2 issect.g . . 3  |-  ( ph  ->  G  e.  ( Y H X ) )
31, 2jca 532 . 2  |-  ( ph  ->  ( F  e.  ( X H Y )  /\  G  e.  ( Y H X ) ) )
4 issect.b . . . . 5  |-  B  =  ( Base `  C
)
5 issect.h . . . . 5  |-  H  =  ( Hom  `  C
)
6 issect.o . . . . 5  |-  .x.  =  (comp `  C )
7 issect.i . . . . 5  |-  .1.  =  ( Id `  C )
8 issect.s . . . . 5  |-  S  =  (Sect `  C )
9 issect.c . . . . 5  |-  ( ph  ->  C  e.  Cat )
10 issect.x . . . . 5  |-  ( ph  ->  X  e.  B )
11 issect.y . . . . 5  |-  ( ph  ->  Y  e.  B )
124, 5, 6, 7, 8, 9, 10, 11issect 14704 . . . 4  |-  ( 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 967 . . . 4  |-  ( ( 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 )
) )
1412, 13syl6bb 261 . . 3  |-  ( ph  ->  ( F ( X S Y ) G  <-> 
( ( F  e.  ( X H Y )  /\  G  e.  ( Y H X ) )  /\  ( G ( <. X ,  Y >.  .x.  X ) F )  =  (  .1.  `  X )
) ) )
1514baibd 900 . 2  |-  ( (
ph  /\  ( F  e.  ( X H Y )  /\  G  e.  ( Y H X ) ) )  -> 
( F ( X S Y ) G  <-> 
( G ( <. X ,  Y >.  .x. 
X ) F )  =  (  .1.  `  X ) ) )
163, 15mpdan 668 1  |-  ( ph  ->  ( F ( X S Y ) G  <-> 
( G ( <. X ,  Y >.  .x. 
X ) F )  =  (  .1.  `  X ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756   <.cop 3895   class class class wbr 4304   ` cfv 5430  (class class class)co 6103   Basecbs 14186   Hom chom 14261  compcco 14262   Catccat 14614   Idccid 14615  Sectcsect 14695
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 4415  ax-sep 4425  ax-nul 4433  ax-pow 4482  ax-pr 4543  ax-un 6384
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 2620  df-ral 2732  df-rex 2733  df-reu 2734  df-rab 2736  df-v 2986  df-sbc 3199  df-csb 3301  df-dif 3343  df-un 3345  df-in 3347  df-ss 3354  df-nul 3650  df-if 3804  df-pw 3874  df-sn 3890  df-pr 3892  df-op 3896  df-uni 4104  df-iun 4185  df-br 4305  df-opab 4363  df-mpt 4364  df-id 4648  df-xp 4858  df-rel 4859  df-cnv 4860  df-co 4861  df-dm 4862  df-rn 4863  df-res 4864  df-ima 4865  df-iota 5393  df-fun 5432  df-fn 5433  df-f 5434  df-f1 5435  df-fo 5436  df-f1o 5437  df-fv 5438  df-ov 6106  df-oprab 6107  df-mpt2 6108  df-1st 6589  df-2nd 6590  df-sect 14698
This theorem is referenced by:  sectco  14707  monsect  14729  funcsect  14794  fthsect  14847  fucsect  14894  catcisolem  14986
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