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Theorem issect2 13972
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 519 . 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 13971 . . . 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 938 . . . 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 253 . . 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 876 . 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 650 1  |-  ( ph  ->  ( F ( X S Y ) G  <-> 
( G ( <. X ,  Y >.  .x. 
X ) F )  =  (  .1.  `  X ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725   <.cop 3809   class class class wbr 4204   ` cfv 5446  (class class class)co 6073   Basecbs 13461    Hom chom 13532  compcco 13533   Catccat 13881   Idccid 13882  Sectcsect 13962
This theorem is referenced by:  sectco  13974  monsect  13996  funcsect  14061  fthsect  14114  fucsect  14161  catcisolem  14253
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-sect 13965
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