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Theorem issect 15130
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 15128 . . 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 4448 . 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 6290 . . . . . 6  |-  ( ( g  =  G  /\  f  =  F )  ->  ( g ( <. X ,  Y >.  .x. 
X ) f )  =  ( G (
<. X ,  Y >.  .x. 
X ) F ) )
1211ancoms 453 . . . . 5  |-  ( ( f  =  F  /\  g  =  G )  ->  ( g ( <. X ,  Y >.  .x. 
X ) f )  =  ( G (
<. X ,  Y >.  .x. 
X ) F ) )
1312eqeq1d 2445 . . . 4  |-  ( ( f  =  F  /\  g  =  G )  ->  ( ( g (
<. X ,  Y >.  .x. 
X ) f )  =  (  .1.  `  X )  <->  ( G
( <. X ,  Y >.  .x.  X ) F )  =  (  .1.  `  X ) ) )
14 eqid 2443 . . . 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 5039 . . 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 369    /\ w3a 974    = wceq 1383    e. wcel 1804   <.cop 4020   class class class wbr 4437   {copab 4494   ` cfv 5578  (class class class)co 6281   Basecbs 14614   Hom chom 14690  compcco 14691   Catccat 15043   Idccid 15044  Sectcsect 15121
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-rep 4548  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676  ax-un 6577
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-ral 2798  df-rex 2799  df-reu 2800  df-rab 2802  df-v 3097  df-sbc 3314  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3771  df-if 3927  df-pw 3999  df-sn 4015  df-pr 4017  df-op 4021  df-uni 4235  df-iun 4317  df-br 4438  df-opab 4496  df-mpt 4497  df-id 4785  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002  df-iota 5541  df-fun 5580  df-fn 5581  df-f 5582  df-f1 5583  df-fo 5584  df-f1o 5585  df-fv 5586  df-ov 6284  df-oprab 6285  df-mpt2 6286  df-1st 6785  df-2nd 6786  df-sect 15124
This theorem is referenced by:  issect2  15131  sectcan  15132  sectco  15133  oppcsect  15150  sectmon  15154  monsect  15155  funcsect  15220  fucsect  15320  invfuc  15322  setcsect  15395  catciso  15413  rngcsect  32663  rngcsectOLD  32675  ringcsect  32711  ringcsectOLD  32735
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