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Theorem issect2 13908
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 13907 . . . 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 1649    e. wcel 1717   <.cop 3761   class class class wbr 4154   ` cfv 5395  (class class class)co 6021   Basecbs 13397    Hom chom 13468  compcco 13469   Catccat 13817   Idccid 13818  Sectcsect 13898
This theorem is referenced by:  sectco  13910  monsect  13932  funcsect  13997  fthsect  14050  fucsect  14097  catcisolem  14189
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369  ax-rep 4262  ax-sep 4272  ax-nul 4280  ax-pow 4319  ax-pr 4345  ax-un 4642
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2243  df-mo 2244  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ne 2553  df-ral 2655  df-rex 2656  df-reu 2657  df-rab 2659  df-v 2902  df-sbc 3106  df-csb 3196  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-nul 3573  df-if 3684  df-pw 3745  df-sn 3764  df-pr 3765  df-op 3767  df-uni 3959  df-iun 4038  df-br 4155  df-opab 4209  df-mpt 4210  df-id 4440  df-xp 4825  df-rel 4826  df-cnv 4827  df-co 4828  df-dm 4829  df-rn 4830  df-res 4831  df-ima 4832  df-iota 5359  df-fun 5397  df-fn 5398  df-f 5399  df-f1 5400  df-fo 5401  df-f1o 5402  df-fv 5403  df-ov 6024  df-oprab 6025  df-mpt2 6026  df-1st 6289  df-2nd 6290  df-sect 13901
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