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Theorem issect2 13657
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 518 . 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 13656 . . . 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 936 . . . 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 252 . . 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 875 . 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 649 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 176    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   <.cop 3643   class class class wbr 4023   ` cfv 5255  (class class class)co 5858   Basecbs 13148    Hom chom 13219  compcco 13220   Catccat 13566   Idccid 13567  Sectcsect 13647
This theorem is referenced by:  sectco  13659  monsect  13681  funcsect  13746  fthsect  13799  fucsect  13846  catcisolem  13938
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-sect 13650
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