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Theorem issect2 13673
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 13672 . . . 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 1632    e. wcel 1696   <.cop 3656   class class class wbr 4039   ` cfv 5271  (class class class)co 5874   Basecbs 13164    Hom chom 13235  compcco 13236   Catccat 13582   Idccid 13583  Sectcsect 13663
This theorem is referenced by:  sectco  13675  monsect  13697  funcsect  13762  fthsect  13815  fucsect  13862  catcisolem  13954
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-sect 13666
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