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Theorem issect 15364
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 15362 . . 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 4405 . 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 6286 . . . . . 6  |-  ( ( g  =  G  /\  f  =  F )  ->  ( g ( <. X ,  Y >.  .x. 
X ) f )  =  ( G (
<. X ,  Y >.  .x. 
X ) F ) )
1211ancoms 451 . . . . 5  |-  ( ( f  =  F  /\  g  =  G )  ->  ( g ( <. X ,  Y >.  .x. 
X ) f )  =  ( G (
<. X ,  Y >.  .x. 
X ) F ) )
1312eqeq1d 2404 . . . 4  |-  ( ( f  =  F  /\  g  =  G )  ->  ( ( g (
<. X ,  Y >.  .x. 
X ) f )  =  (  .1.  `  X )  <->  ( G
( <. X ,  Y >.  .x.  X ) F )  =  (  .1.  `  X ) ) )
14 eqid 2402 . . . 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 4872 . . 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 976 . . 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 252 . 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 261 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 setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367    /\ w3a 974    = wceq 1405    e. wcel 1842   <.cop 3977   class class class wbr 4394   {copab 4451   ` cfv 5568  (class class class)co 6277   Basecbs 14839   Hom chom 14918  compcco 14919   Catccat 15276   Idccid 15277  Sectcsect 15355
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4506  ax-sep 4516  ax-nul 4524  ax-pow 4571  ax-pr 4629  ax-un 6573
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2758  df-rex 2759  df-reu 2760  df-rab 2762  df-v 3060  df-sbc 3277  df-csb 3373  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-nul 3738  df-if 3885  df-pw 3956  df-sn 3972  df-pr 3974  df-op 3978  df-uni 4191  df-iun 4272  df-br 4395  df-opab 4453  df-mpt 4454  df-id 4737  df-xp 4828  df-rel 4829  df-cnv 4830  df-co 4831  df-dm 4832  df-rn 4833  df-res 4834  df-ima 4835  df-iota 5532  df-fun 5570  df-fn 5571  df-f 5572  df-f1 5573  df-fo 5574  df-f1o 5575  df-fv 5576  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-1st 6783  df-2nd 6784  df-sect 15358
This theorem is referenced by:  issect2  15365  sectcan  15366  sectco  15367  oppcsect  15389  sectmon  15393  monsect  15394  funcsect  15483  fucsect  15583  invfuc  15585  setcsect  15690  catciso  15708  rngcsect  38280  rngcsectALTV  38292  ringcsect  38331  ringcsectALTV  38355
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