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Theorem sectffval 15165
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 5872 . . . . . 6  |-  ( c  =  C  ->  ( Base `  c )  =  ( Base `  C
) )
4 issect.b . . . . . 6  |-  B  =  ( Base `  C
)
53, 4syl6eqr 2516 . . . . 5  |-  ( c  =  C  ->  ( Base `  c )  =  B )
6 fvex 5882 . . . . . . . 8  |-  ( Hom  `  c )  e.  _V
76a1i 11 . . . . . . 7  |-  ( c  =  C  ->  ( Hom  `  c )  e. 
_V )
8 fveq2 5872 . . . . . . . 8  |-  ( c  =  C  ->  ( Hom  `  c )  =  ( Hom  `  C
) )
9 issect.h . . . . . . . 8  |-  H  =  ( Hom  `  C
)
108, 9syl6eqr 2516 . . . . . . 7  |-  ( c  =  C  ->  ( Hom  `  c )  =  H )
11 simpr 461 . . . . . . . . . . 11  |-  ( ( c  =  C  /\  h  =  H )  ->  h  =  H )
1211oveqd 6313 . . . . . . . . . 10  |-  ( ( c  =  C  /\  h  =  H )  ->  ( x h y )  =  ( x H y ) )
1312eleq2d 2527 . . . . . . . . 9  |-  ( ( c  =  C  /\  h  =  H )  ->  ( f  e.  ( x h y )  <-> 
f  e.  ( x H y ) ) )
1411oveqd 6313 . . . . . . . . . 10  |-  ( ( c  =  C  /\  h  =  H )  ->  ( y h x )  =  ( y H x ) )
1514eleq2d 2527 . . . . . . . . 9  |-  ( ( c  =  C  /\  h  =  H )  ->  ( g  e.  ( y h x )  <-> 
g  e.  ( y H x ) ) )
1613, 15anbi12d 710 . . . . . . . 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 457 . . . . . . . . . . . . 13  |-  ( ( c  =  C  /\  h  =  H )  ->  c  =  C )
1817fveq2d 5876 . . . . . . . . . . . 12  |-  ( ( c  =  C  /\  h  =  H )  ->  (comp `  c )  =  (comp `  C )
)
19 issect.o . . . . . . . . . . . 12  |-  .x.  =  (comp `  C )
2018, 19syl6eqr 2516 . . . . . . . . . . 11  |-  ( ( c  =  C  /\  h  =  H )  ->  (comp `  c )  =  .x.  )
2120oveqd 6313 . . . . . . . . . 10  |-  ( ( c  =  C  /\  h  =  H )  ->  ( <. x ,  y
>. (comp `  c )
x )  =  (
<. x ,  y >.  .x.  x ) )
2221oveqd 6313 . . . . . . . . 9  |-  ( ( c  =  C  /\  h  =  H )  ->  ( g ( <.
x ,  y >.
(comp `  c )
x ) f )  =  ( g (
<. x ,  y >.  .x.  x ) f ) )
2317fveq2d 5876 . . . . . . . . . . 11  |-  ( ( c  =  C  /\  h  =  H )  ->  ( Id `  c
)  =  ( Id
`  C ) )
24 issect.i . . . . . . . . . . 11  |-  .1.  =  ( Id `  C )
2523, 24syl6eqr 2516 . . . . . . . . . 10  |-  ( ( c  =  C  /\  h  =  H )  ->  ( Id `  c
)  =  .1.  )
2625fveq1d 5874 . . . . . . . . 9  |-  ( ( c  =  C  /\  h  =  H )  ->  ( ( Id `  c ) `  x
)  =  (  .1.  `  x ) )
2722, 26eqeq12d 2479 . . . . . . . 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 710 . . . . . . 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 3365 . . . . . 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 4520 . . . . 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 6358 . . . 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 15162 . . . 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 5882 . . . . . 6  |-  ( Base `  C )  e.  _V
344, 33eqeltri 2541 . . . . 5  |-  B  e. 
_V
3534, 34mpt2ex 6876 . . . 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 5956 . . 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 2510 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 setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1395    e. wcel 1819   _Vcvv 3109   [.wsbc 3327   <.cop 4038   {copab 4514   ` cfv 5594  (class class class)co 6296    |-> cmpt2 6298   Basecbs 14643   Hom chom 14722  compcco 14723   Catccat 15080   Idccid 15081  Sectcsect 15159
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-id 4804  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-1st 6799  df-2nd 6800  df-sect 15162
This theorem is referenced by:  sectfval  15166
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