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Theorem sectffval 13669
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 5541 . . . . . 6  |-  ( c  =  C  ->  ( Base `  c )  =  ( Base `  C
) )
4 issect.b . . . . . 6  |-  B  =  ( Base `  C
)
53, 4syl6eqr 2346 . . . . 5  |-  ( c  =  C  ->  ( Base `  c )  =  B )
6 fvex 5555 . . . . . . . 8  |-  (  Hom  `  c )  e.  _V
76a1i 10 . . . . . . 7  |-  ( c  =  C  ->  (  Hom  `  c )  e. 
_V )
8 fveq2 5541 . . . . . . . 8  |-  ( c  =  C  ->  (  Hom  `  c )  =  (  Hom  `  C
) )
9 issect.h . . . . . . . 8  |-  H  =  (  Hom  `  C
)
108, 9syl6eqr 2346 . . . . . . 7  |-  ( c  =  C  ->  (  Hom  `  c )  =  H )
11 simpr 447 . . . . . . . . . . 11  |-  ( ( c  =  C  /\  h  =  H )  ->  h  =  H )
1211oveqd 5891 . . . . . . . . . 10  |-  ( ( c  =  C  /\  h  =  H )  ->  ( x h y )  =  ( x H y ) )
1312eleq2d 2363 . . . . . . . . 9  |-  ( ( c  =  C  /\  h  =  H )  ->  ( f  e.  ( x h y )  <-> 
f  e.  ( x H y ) ) )
1411oveqd 5891 . . . . . . . . . 10  |-  ( ( c  =  C  /\  h  =  H )  ->  ( y h x )  =  ( y H x ) )
1514eleq2d 2363 . . . . . . . . 9  |-  ( ( c  =  C  /\  h  =  H )  ->  ( g  e.  ( y h x )  <-> 
g  e.  ( y H x ) ) )
1613, 15anbi12d 691 . . . . . . . 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 443 . . . . . . . . . . . . 13  |-  ( ( c  =  C  /\  h  =  H )  ->  c  =  C )
1817fveq2d 5545 . . . . . . . . . . . 12  |-  ( ( c  =  C  /\  h  =  H )  ->  (comp `  c )  =  (comp `  C )
)
19 issect.o . . . . . . . . . . . 12  |-  .x.  =  (comp `  C )
2018, 19syl6eqr 2346 . . . . . . . . . . 11  |-  ( ( c  =  C  /\  h  =  H )  ->  (comp `  c )  =  .x.  )
2120oveqd 5891 . . . . . . . . . 10  |-  ( ( c  =  C  /\  h  =  H )  ->  ( <. x ,  y
>. (comp `  c )
x )  =  (
<. x ,  y >.  .x.  x ) )
2221oveqd 5891 . . . . . . . . 9  |-  ( ( c  =  C  /\  h  =  H )  ->  ( g ( <.
x ,  y >.
(comp `  c )
x ) f )  =  ( g (
<. x ,  y >.  .x.  x ) f ) )
2317fveq2d 5545 . . . . . . . . . . 11  |-  ( ( c  =  C  /\  h  =  H )  ->  ( Id `  c
)  =  ( Id
`  C ) )
24 issect.i . . . . . . . . . . 11  |-  .1.  =  ( Id `  C )
2523, 24syl6eqr 2346 . . . . . . . . . 10  |-  ( ( c  =  C  /\  h  =  H )  ->  ( Id `  c
)  =  .1.  )
2625fveq1d 5543 . . . . . . . . 9  |-  ( ( c  =  C  /\  h  =  H )  ->  ( ( Id `  c ) `  x
)  =  (  .1.  `  x ) )
2722, 26eqeq12d 2310 . . . . . . . 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 691 . . . . . . 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 3041 . . . . . 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 4098 . . . . 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 5926 . . . 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 13666 . . . 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 5555 . . . . . 6  |-  ( Base `  C )  e.  _V
344, 33eqeltri 2366 . . . . 5  |-  B  e. 
_V
3534, 34mpt2ex 6214 . . . 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 5618 . . 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 15 . 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 2340 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 358    = wceq 1632    e. wcel 1696   _Vcvv 2801   [.wsbc 3004   <.cop 3656   {copab 4092   ` cfv 5271  (class class class)co 5874    e. cmpt2 5876   Basecbs 13164    Hom chom 13235  compcco 13236   Catccat 13582   Idccid 13583  Sectcsect 13663
This theorem is referenced by:  sectfval  13670
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-sect 13666
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