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Theorem issect 15658
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 15656 . . 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 4413 . 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 6299 . . . . . 6  |-  ( ( g  =  G  /\  f  =  F )  ->  ( g ( <. X ,  Y >.  .x. 
X ) f )  =  ( G (
<. X ,  Y >.  .x. 
X ) F ) )
1211ancoms 455 . . . . 5  |-  ( ( f  =  F  /\  g  =  G )  ->  ( g ( <. X ,  Y >.  .x. 
X ) f )  =  ( G (
<. X ,  Y >.  .x. 
X ) F ) )
1312eqeq1d 2453 . . . 4  |-  ( ( f  =  F  /\  g  =  G )  ->  ( ( g (
<. X ,  Y >.  .x. 
X ) f )  =  (  .1.  `  X )  <->  ( G
( <. X ,  Y >.  .x.  X ) F )  =  (  .1.  `  X ) ) )
14 eqid 2451 . . . 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 4884 . . 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 987 . . 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 256 . 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 265 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 188    /\ wa 371    /\ w3a 985    = wceq 1444    e. wcel 1887   <.cop 3974   class class class wbr 4402   {copab 4460   ` cfv 5582  (class class class)co 6290   Basecbs 15121   Hom chom 15201  compcco 15202   Catccat 15570   Idccid 15571  Sectcsect 15649
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-rep 4515  ax-sep 4525  ax-nul 4534  ax-pow 4581  ax-pr 4639  ax-un 6583
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-ral 2742  df-rex 2743  df-reu 2744  df-rab 2746  df-v 3047  df-sbc 3268  df-csb 3364  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-nul 3732  df-if 3882  df-pw 3953  df-sn 3969  df-pr 3971  df-op 3975  df-uni 4199  df-iun 4280  df-br 4403  df-opab 4462  df-mpt 4463  df-id 4749  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-res 4846  df-ima 4847  df-iota 5546  df-fun 5584  df-fn 5585  df-f 5586  df-f1 5587  df-fo 5588  df-f1o 5589  df-fv 5590  df-ov 6293  df-oprab 6294  df-mpt2 6295  df-1st 6793  df-2nd 6794  df-sect 15652
This theorem is referenced by:  issect2  15659  sectcan  15660  sectco  15661  oppcsect  15683  sectmon  15687  monsect  15688  funcsect  15777  fucsect  15877  invfuc  15879  setcsect  15984  catciso  16002  rngcsect  40035  rngcsectALTV  40047  ringcsect  40086  ringcsectALTV  40110
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