MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  sectffval Structured version   Unicode version

Theorem sectffval 14689
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 5691 . . . . . 6  |-  ( c  =  C  ->  ( Base `  c )  =  ( Base `  C
) )
4 issect.b . . . . . 6  |-  B  =  ( Base `  C
)
53, 4syl6eqr 2493 . . . . 5  |-  ( c  =  C  ->  ( Base `  c )  =  B )
6 fvex 5701 . . . . . . . 8  |-  ( Hom  `  c )  e.  _V
76a1i 11 . . . . . . 7  |-  ( c  =  C  ->  ( Hom  `  c )  e. 
_V )
8 fveq2 5691 . . . . . . . 8  |-  ( c  =  C  ->  ( Hom  `  c )  =  ( Hom  `  C
) )
9 issect.h . . . . . . . 8  |-  H  =  ( Hom  `  C
)
108, 9syl6eqr 2493 . . . . . . 7  |-  ( c  =  C  ->  ( Hom  `  c )  =  H )
11 simpr 461 . . . . . . . . . . 11  |-  ( ( c  =  C  /\  h  =  H )  ->  h  =  H )
1211oveqd 6108 . . . . . . . . . 10  |-  ( ( c  =  C  /\  h  =  H )  ->  ( x h y )  =  ( x H y ) )
1312eleq2d 2510 . . . . . . . . 9  |-  ( ( c  =  C  /\  h  =  H )  ->  ( f  e.  ( x h y )  <-> 
f  e.  ( x H y ) ) )
1411oveqd 6108 . . . . . . . . . 10  |-  ( ( c  =  C  /\  h  =  H )  ->  ( y h x )  =  ( y H x ) )
1514eleq2d 2510 . . . . . . . . 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 5695 . . . . . . . . . . . 12  |-  ( ( c  =  C  /\  h  =  H )  ->  (comp `  c )  =  (comp `  C )
)
19 issect.o . . . . . . . . . . . 12  |-  .x.  =  (comp `  C )
2018, 19syl6eqr 2493 . . . . . . . . . . 11  |-  ( ( c  =  C  /\  h  =  H )  ->  (comp `  c )  =  .x.  )
2120oveqd 6108 . . . . . . . . . 10  |-  ( ( c  =  C  /\  h  =  H )  ->  ( <. x ,  y
>. (comp `  c )
x )  =  (
<. x ,  y >.  .x.  x ) )
2221oveqd 6108 . . . . . . . . 9  |-  ( ( c  =  C  /\  h  =  H )  ->  ( g ( <.
x ,  y >.
(comp `  c )
x ) f )  =  ( g (
<. x ,  y >.  .x.  x ) f ) )
2317fveq2d 5695 . . . . . . . . . . 11  |-  ( ( c  =  C  /\  h  =  H )  ->  ( Id `  c
)  =  ( Id
`  C ) )
24 issect.i . . . . . . . . . . 11  |-  .1.  =  ( Id `  C )
2523, 24syl6eqr 2493 . . . . . . . . . 10  |-  ( ( c  =  C  /\  h  =  H )  ->  ( Id `  c
)  =  .1.  )
2625fveq1d 5693 . . . . . . . . 9  |-  ( ( c  =  C  /\  h  =  H )  ->  ( ( Id `  c ) `  x
)  =  (  .1.  `  x ) )
2722, 26eqeq12d 2457 . . . . . . . 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 3224 . . . . . 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 4355 . . . . 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 6148 . . . 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 14686 . . . 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 5701 . . . . . 6  |-  ( Base `  C )  e.  _V
344, 33eqeltri 2513 . . . . 5  |-  B  e. 
_V
3534, 34mpt2ex 6650 . . . 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 5774 . . 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 2487 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 1369    e. wcel 1756   _Vcvv 2972   [.wsbc 3186   <.cop 3883   {copab 4349   ` cfv 5418  (class class class)co 6091    e. cmpt2 6093   Basecbs 14174   Hom chom 14249  compcco 14250   Catccat 14602   Idccid 14603  Sectcsect 14683
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4403  ax-sep 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531  ax-un 6372
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-ral 2720  df-rex 2721  df-reu 2722  df-rab 2724  df-v 2974  df-sbc 3187  df-csb 3289  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-nul 3638  df-if 3792  df-pw 3862  df-sn 3878  df-pr 3880  df-op 3884  df-uni 4092  df-iun 4173  df-br 4293  df-opab 4351  df-mpt 4352  df-id 4636  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-ima 4853  df-iota 5381  df-fun 5420  df-fn 5421  df-f 5422  df-f1 5423  df-fo 5424  df-f1o 5425  df-fv 5426  df-ov 6094  df-oprab 6095  df-mpt2 6096  df-1st 6577  df-2nd 6578  df-sect 14686
This theorem is referenced by:  sectfval  14690
  Copyright terms: Public domain W3C validator