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Theorem issect 13934
Description: The property " F is a section of  G". (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 )
Assertion
Ref Expression
issect  |-  ( ph  ->  ( F ( X S Y ) G  <-> 
( F  e.  ( X H Y )  /\  G  e.  ( Y H X )  /\  ( G (
<. X ,  Y >.  .x. 
X ) F )  =  (  .1.  `  X ) ) ) )

Proof of Theorem issect
Dummy variables  f 
g are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 issect.b . . . 4  |-  B  =  ( Base `  C
)
2 issect.h . . . 4  |-  H  =  (  Hom  `  C
)
3 issect.o . . . 4  |-  .x.  =  (comp `  C )
4 issect.i . . . 4  |-  .1.  =  ( Id `  C )
5 issect.s . . . 4  |-  S  =  (Sect `  C )
6 issect.c . . . 4  |-  ( ph  ->  C  e.  Cat )
7 issect.x . . . 4  |-  ( ph  ->  X  e.  B )
8 issect.y . . . 4  |-  ( ph  ->  Y  e.  B )
91, 2, 3, 4, 5, 6, 7, 8sectfval 13932 . . 3  |-  ( ph  ->  ( X S Y )  =  { <. f ,  g >.  |  ( ( f  e.  ( X H Y )  /\  g  e.  ( Y H X ) )  /\  ( g ( <. X ,  Y >.  .x.  X ) f )  =  (  .1.  `  X ) ) } )
109breqd 4183 . 2  |-  ( ph  ->  ( F ( X S Y ) G  <-> 
F { <. f ,  g >.  |  ( ( f  e.  ( X H Y )  /\  g  e.  ( Y H X ) )  /\  ( g ( <. X ,  Y >.  .x.  X ) f )  =  (  .1.  `  X ) ) } G ) )
11 oveq12 6049 . . . . . 6  |-  ( ( g  =  G  /\  f  =  F )  ->  ( g ( <. X ,  Y >.  .x. 
X ) f )  =  ( G (
<. X ,  Y >.  .x. 
X ) F ) )
1211ancoms 440 . . . . 5  |-  ( ( f  =  F  /\  g  =  G )  ->  ( g ( <. X ,  Y >.  .x. 
X ) f )  =  ( G (
<. X ,  Y >.  .x. 
X ) F ) )
1312eqeq1d 2412 . . . 4  |-  ( ( f  =  F  /\  g  =  G )  ->  ( ( g (
<. X ,  Y >.  .x. 
X ) f )  =  (  .1.  `  X )  <->  ( G
( <. X ,  Y >.  .x.  X ) F )  =  (  .1.  `  X ) ) )
14 eqid 2404 . . . 4  |-  { <. f ,  g >.  |  ( ( f  e.  ( X H Y )  /\  g  e.  ( Y H X ) )  /\  ( g ( <. X ,  Y >.  .x.  X ) f )  =  (  .1.  `  X ) ) }  =  { <. f ,  g >.  |  ( ( f  e.  ( X H Y )  /\  g  e.  ( Y H X ) )  /\  ( g ( <. X ,  Y >.  .x.  X ) f )  =  (  .1.  `  X ) ) }
1513, 14brab2a 4886 . . 3  |-  ( F { <. f ,  g
>.  |  ( (
f  e.  ( X H Y )  /\  g  e.  ( Y H X ) )  /\  ( g ( <. X ,  Y >.  .x. 
X ) f )  =  (  .1.  `  X ) ) } G  <->  ( ( F  e.  ( X H Y )  /\  G  e.  ( Y H X ) )  /\  ( G ( <. X ,  Y >.  .x.  X ) F )  =  (  .1.  `  X )
) )
16 df-3an 938 . . 3  |-  ( ( 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 )
) )
1715, 16bitr4i 244 . 2  |-  ( F { <. f ,  g
>.  |  ( (
f  e.  ( X H Y )  /\  g  e.  ( Y H X ) )  /\  ( g ( <. X ,  Y >.  .x. 
X ) f )  =  (  .1.  `  X ) ) } G  <->  ( F  e.  ( X H Y )  /\  G  e.  ( Y H X )  /\  ( G ( <. X ,  Y >.  .x.  X ) F )  =  (  .1.  `  X ) ) )
1810, 17syl6bb 253 1  |-  ( ph  ->  ( F ( X S Y ) G  <-> 
( F  e.  ( X H Y )  /\  G  e.  ( Y H X )  /\  ( 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 1721   <.cop 3777   class class class wbr 4172   {copab 4225   ` cfv 5413  (class class class)co 6040   Basecbs 13424    Hom chom 13495  compcco 13496   Catccat 13844   Idccid 13845  Sectcsect 13925
This theorem is referenced by:  issect2  13935  sectcan  13936  sectco  13937  oppcsect  13954  sectmon  13958  monsect  13959  funcsect  14024  fucsect  14124  invfuc  14126  setcsect  14199  catciso  14217
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 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660
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 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-reu 2673  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-id 4458  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-1st 6308  df-2nd 6309  df-sect 13928
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