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Theorem sectffval 13903
Description: Value of the section operation. (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
sectffval  |-  ( ph  ->  S  =  ( x  e.  B ,  y  e.  B  |->  { <. f ,  g >.  |  ( ( f  e.  ( x H y )  /\  g  e.  ( y H x ) )  /\  ( g ( <. x ,  y
>.  .x.  x ) f )  =  (  .1.  `  x ) ) } ) )
Distinct variable groups:    f, g, x, y,  .1.    x, B, y    C, f, g, x, y    ph, f, g, x, y    f, H, g, x, y    .x. , f,
g, x, y    f, X, g, x, y    f, Y, g, x, y
Allowed substitution hints:    B( f, g)    S( x, y, f, g)

Proof of Theorem sectffval
Dummy variables  c  h are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 issect.s . 2  |-  S  =  (Sect `  C )
2 issect.c . . 3  |-  ( ph  ->  C  e.  Cat )
3 fveq2 5668 . . . . . 6  |-  ( c  =  C  ->  ( Base `  c )  =  ( Base `  C
) )
4 issect.b . . . . . 6  |-  B  =  ( Base `  C
)
53, 4syl6eqr 2437 . . . . 5  |-  ( c  =  C  ->  ( Base `  c )  =  B )
6 fvex 5682 . . . . . . . 8  |-  (  Hom  `  c )  e.  _V
76a1i 11 . . . . . . 7  |-  ( c  =  C  ->  (  Hom  `  c )  e. 
_V )
8 fveq2 5668 . . . . . . . 8  |-  ( c  =  C  ->  (  Hom  `  c )  =  (  Hom  `  C
) )
9 issect.h . . . . . . . 8  |-  H  =  (  Hom  `  C
)
108, 9syl6eqr 2437 . . . . . . 7  |-  ( c  =  C  ->  (  Hom  `  c )  =  H )
11 simpr 448 . . . . . . . . . . 11  |-  ( ( c  =  C  /\  h  =  H )  ->  h  =  H )
1211oveqd 6037 . . . . . . . . . 10  |-  ( ( c  =  C  /\  h  =  H )  ->  ( x h y )  =  ( x H y ) )
1312eleq2d 2454 . . . . . . . . 9  |-  ( ( c  =  C  /\  h  =  H )  ->  ( f  e.  ( x h y )  <-> 
f  e.  ( x H y ) ) )
1411oveqd 6037 . . . . . . . . . 10  |-  ( ( c  =  C  /\  h  =  H )  ->  ( y h x )  =  ( y H x ) )
1514eleq2d 2454 . . . . . . . . 9  |-  ( ( c  =  C  /\  h  =  H )  ->  ( g  e.  ( y h x )  <-> 
g  e.  ( y H x ) ) )
1613, 15anbi12d 692 . . . . . . . 8  |-  ( ( c  =  C  /\  h  =  H )  ->  ( ( f  e.  ( x h y )  /\  g  e.  ( y h x ) )  <->  ( f  e.  ( x H y )  /\  g  e.  ( y H x ) ) ) )
17 simpl 444 . . . . . . . . . . . . 13  |-  ( ( c  =  C  /\  h  =  H )  ->  c  =  C )
1817fveq2d 5672 . . . . . . . . . . . 12  |-  ( ( c  =  C  /\  h  =  H )  ->  (comp `  c )  =  (comp `  C )
)
19 issect.o . . . . . . . . . . . 12  |-  .x.  =  (comp `  C )
2018, 19syl6eqr 2437 . . . . . . . . . . 11  |-  ( ( c  =  C  /\  h  =  H )  ->  (comp `  c )  =  .x.  )
2120oveqd 6037 . . . . . . . . . 10  |-  ( ( c  =  C  /\  h  =  H )  ->  ( <. x ,  y
>. (comp `  c )
x )  =  (
<. x ,  y >.  .x.  x ) )
2221oveqd 6037 . . . . . . . . 9  |-  ( ( c  =  C  /\  h  =  H )  ->  ( g ( <.
x ,  y >.
(comp `  c )
x ) f )  =  ( g (
<. x ,  y >.  .x.  x ) f ) )
2317fveq2d 5672 . . . . . . . . . . 11  |-  ( ( c  =  C  /\  h  =  H )  ->  ( Id `  c
)  =  ( Id
`  C ) )
24 issect.i . . . . . . . . . . 11  |-  .1.  =  ( Id `  C )
2523, 24syl6eqr 2437 . . . . . . . . . 10  |-  ( ( c  =  C  /\  h  =  H )  ->  ( Id `  c
)  =  .1.  )
2625fveq1d 5670 . . . . . . . . 9  |-  ( ( c  =  C  /\  h  =  H )  ->  ( ( Id `  c ) `  x
)  =  (  .1.  `  x ) )
2722, 26eqeq12d 2401 . . . . . . . 8  |-  ( ( c  =  C  /\  h  =  H )  ->  ( ( g (
<. x ,  y >.
(comp `  c )
x ) f )  =  ( ( Id
`  c ) `  x )  <->  ( g
( <. x ,  y
>.  .x.  x ) f )  =  (  .1.  `  x ) ) )
2816, 27anbi12d 692 . . . . . . 7  |-  ( ( c  =  C  /\  h  =  H )  ->  ( ( ( f  e.  ( x h y )  /\  g  e.  ( y h x ) )  /\  (
g ( <. x ,  y >. (comp `  c ) x ) f )  =  ( ( Id `  c
) `  x )
)  <->  ( ( f  e.  ( x H y )  /\  g  e.  ( y H x ) )  /\  (
g ( <. x ,  y >.  .x.  x
) f )  =  (  .1.  `  x
) ) ) )
297, 10, 28sbcied2 3141 . . . . . 6  |-  ( c  =  C  ->  ( [. (  Hom  `  c
)  /  h ]. ( ( f  e.  ( x h y )  /\  g  e.  ( y h x ) )  /\  (
g ( <. x ,  y >. (comp `  c ) x ) f )  =  ( ( Id `  c
) `  x )
)  <->  ( ( f  e.  ( x H y )  /\  g  e.  ( y H x ) )  /\  (
g ( <. x ,  y >.  .x.  x
) f )  =  (  .1.  `  x
) ) ) )
3029opabbidv 4212 . . . . 5  |-  ( c  =  C  ->  { <. f ,  g >.  |  [. (  Hom  `  c )  /  h ]. ( ( f  e.  ( x h y )  /\  g  e.  ( y
h x ) )  /\  ( g (
<. x ,  y >.
(comp `  c )
x ) f )  =  ( ( Id
`  c ) `  x ) ) }  =  { <. f ,  g >.  |  ( ( f  e.  ( x H y )  /\  g  e.  ( y H x ) )  /\  ( g ( <. x ,  y
>.  .x.  x ) f )  =  (  .1.  `  x ) ) } )
315, 5, 30mpt2eq123dv 6075 . . . 4  |-  ( c  =  C  ->  (
x  e.  ( Base `  c ) ,  y  e.  ( Base `  c
)  |->  { <. f ,  g >.  |  [. (  Hom  `  c )  /  h ]. ( ( f  e.  ( x h y )  /\  g  e.  ( y
h x ) )  /\  ( g (
<. x ,  y >.
(comp `  c )
x ) f )  =  ( ( Id
`  c ) `  x ) ) } )  =  ( x  e.  B ,  y  e.  B  |->  { <. f ,  g >.  |  ( ( f  e.  ( x H y )  /\  g  e.  ( y H x ) )  /\  ( g ( <. x ,  y
>.  .x.  x ) f )  =  (  .1.  `  x ) ) } ) )
32 df-sect 13900 . . . 4  |- Sect  =  ( c  e.  Cat  |->  ( x  e.  ( Base `  c ) ,  y  e.  ( Base `  c
)  |->  { <. f ,  g >.  |  [. (  Hom  `  c )  /  h ]. ( ( f  e.  ( x h y )  /\  g  e.  ( y
h x ) )  /\  ( g (
<. x ,  y >.
(comp `  c )
x ) f )  =  ( ( Id
`  c ) `  x ) ) } ) )
33 fvex 5682 . . . . . 6  |-  ( Base `  C )  e.  _V
344, 33eqeltri 2457 . . . . 5  |-  B  e. 
_V
3534, 34mpt2ex 6364 . . . 4  |-  ( x  e.  B ,  y  e.  B  |->  { <. f ,  g >.  |  ( ( f  e.  ( x H y )  /\  g  e.  ( y H x ) )  /\  ( g ( <. x ,  y
>.  .x.  x ) f )  =  (  .1.  `  x ) ) } )  e.  _V
3631, 32, 35fvmpt 5745 . . 3  |-  ( C  e.  Cat  ->  (Sect `  C )  =  ( x  e.  B , 
y  e.  B  |->  {
<. f ,  g >.  |  ( ( f  e.  ( x H y )  /\  g  e.  ( y H x ) )  /\  (
g ( <. x ,  y >.  .x.  x
) f )  =  (  .1.  `  x
) ) } ) )
372, 36syl 16 . 2  |-  ( ph  ->  (Sect `  C )  =  ( x  e.  B ,  y  e.  B  |->  { <. f ,  g >.  |  ( ( f  e.  ( x H y )  /\  g  e.  ( y H x ) )  /\  ( g ( <. x ,  y
>.  .x.  x ) f )  =  (  .1.  `  x ) ) } ) )
381, 37syl5eq 2431 1  |-  ( ph  ->  S  =  ( x  e.  B ,  y  e.  B  |->  { <. f ,  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    /\ wa 359    = wceq 1649    e. wcel 1717   _Vcvv 2899   [.wsbc 3104   <.cop 3760   {copab 4206   ` cfv 5394  (class class class)co 6020    e. cmpt2 6022   Basecbs 13396    Hom chom 13467  compcco 13468   Catccat 13816   Idccid 13817  Sectcsect 13897
This theorem is referenced by:  sectfval  13904
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 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-rep 4261  ax-sep 4271  ax-nul 4279  ax-pow 4318  ax-pr 4344  ax-un 4641
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 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-ral 2654  df-rex 2655  df-reu 2656  df-rab 2658  df-v 2901  df-sbc 3105  df-csb 3195  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-nul 3572  df-if 3683  df-pw 3744  df-sn 3763  df-pr 3764  df-op 3766  df-uni 3958  df-iun 4037  df-br 4154  df-opab 4208  df-mpt 4209  df-id 4439  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-rn 4829  df-res 4830  df-ima 4831  df-iota 5358  df-fun 5396  df-fn 5397  df-f 5398  df-f1 5399  df-fo 5400  df-f1o 5401  df-fv 5402  df-ov 6023  df-oprab 6024  df-mpt2 6025  df-1st 6288  df-2nd 6289  df-sect 13900
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