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Theorem sectfval 13977
Description: Value of the section relation. (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
sectfval  |-  ( ph  ->  ( X S Y )  =  { <. 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,  .1.    C, f, g    ph, f,
g    f, H, g    .x. , f,
g    f, X, g    f, Y, g
Allowed substitution hints:    B( f, g)    S( f, g)

Proof of Theorem sectfval
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 issect.b . . 3  |-  B  =  ( Base `  C
)
2 issect.h . . 3  |-  H  =  (  Hom  `  C
)
3 issect.o . . 3  |-  .x.  =  (comp `  C )
4 issect.i . . 3  |-  .1.  =  ( Id `  C )
5 issect.s . . 3  |-  S  =  (Sect `  C )
6 issect.c . . 3  |-  ( ph  ->  C  e.  Cat )
7 issect.x . . 3  |-  ( ph  ->  X  e.  B )
81, 2, 3, 4, 5, 6, 7, 7sectffval 13976 . 2  |-  ( 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 ) ) } ) )
9 simprl 733 . . . . . . 7  |-  ( (
ph  /\  ( x  =  X  /\  y  =  Y ) )  ->  x  =  X )
10 simprr 734 . . . . . . 7  |-  ( (
ph  /\  ( x  =  X  /\  y  =  Y ) )  -> 
y  =  Y )
119, 10oveq12d 6099 . . . . . 6  |-  ( (
ph  /\  ( x  =  X  /\  y  =  Y ) )  -> 
( x H y )  =  ( X H Y ) )
1211eleq2d 2503 . . . . 5  |-  ( (
ph  /\  ( x  =  X  /\  y  =  Y ) )  -> 
( f  e.  ( x H y )  <-> 
f  e.  ( X H Y ) ) )
1310, 9oveq12d 6099 . . . . . 6  |-  ( (
ph  /\  ( x  =  X  /\  y  =  Y ) )  -> 
( y H x )  =  ( Y H X ) )
1413eleq2d 2503 . . . . 5  |-  ( (
ph  /\  ( x  =  X  /\  y  =  Y ) )  -> 
( g  e.  ( y H x )  <-> 
g  e.  ( Y H X ) ) )
1512, 14anbi12d 692 . . . 4  |-  ( (
ph  /\  ( x  =  X  /\  y  =  Y ) )  -> 
( ( f  e.  ( x H y )  /\  g  e.  ( y H x ) )  <->  ( f  e.  ( X H Y )  /\  g  e.  ( Y H X ) ) ) )
169, 10opeq12d 3992 . . . . . . 7  |-  ( (
ph  /\  ( x  =  X  /\  y  =  Y ) )  ->  <. x ,  y >.  =  <. X ,  Y >. )
1716, 9oveq12d 6099 . . . . . 6  |-  ( (
ph  /\  ( x  =  X  /\  y  =  Y ) )  -> 
( <. x ,  y
>.  .x.  x )  =  ( <. X ,  Y >.  .x.  X ) )
1817oveqd 6098 . . . . 5  |-  ( (
ph  /\  ( x  =  X  /\  y  =  Y ) )  -> 
( g ( <.
x ,  y >.  .x.  x ) f )  =  ( g (
<. X ,  Y >.  .x. 
X ) f ) )
199fveq2d 5732 . . . . 5  |-  ( (
ph  /\  ( x  =  X  /\  y  =  Y ) )  -> 
(  .1.  `  x
)  =  (  .1.  `  X ) )
2018, 19eqeq12d 2450 . . . 4  |-  ( (
ph  /\  ( x  =  X  /\  y  =  Y ) )  -> 
( ( g (
<. x ,  y >.  .x.  x ) f )  =  (  .1.  `  x )  <->  ( g
( <. X ,  Y >.  .x.  X ) f )  =  (  .1.  `  X ) ) )
2115, 20anbi12d 692 . . 3  |-  ( (
ph  /\  ( x  =  X  /\  y  =  Y ) )  -> 
( ( ( 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 ) ) ) )
2221opabbidv 4271 . 2  |-  ( (
ph  /\  ( x  =  X  /\  y  =  Y ) )  ->  { <. 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 ) ) } )
23 issect.y . 2  |-  ( ph  ->  Y  e.  B )
24 ovex 6106 . . . . 5  |-  ( X H Y )  e. 
_V
25 ovex 6106 . . . . 5  |-  ( Y H X )  e. 
_V
2624, 25xpex 4990 . . . 4  |-  ( ( X H Y )  X.  ( Y H X ) )  e. 
_V
27 opabssxp 4950 . . . 4  |-  { <. f ,  g >.  |  ( ( f  e.  ( X H Y )  /\  g  e.  ( Y H X ) )  /\  ( g ( <. X ,  Y >.  .x.  X ) f )  =  (  .1.  `  X ) ) } 
C_  ( ( X H Y )  X.  ( Y H X ) )
2826, 27ssexi 4348 . . 3  |-  { <. f ,  g >.  |  ( ( f  e.  ( X H Y )  /\  g  e.  ( Y H X ) )  /\  ( g ( <. X ,  Y >.  .x.  X ) f )  =  (  .1.  `  X ) ) }  e.  _V
2928a1i 11 . 2  |-  ( ph  ->  { <. f ,  g
>.  |  ( (
f  e.  ( X H Y )  /\  g  e.  ( Y H X ) )  /\  ( g ( <. X ,  Y >.  .x. 
X ) f )  =  (  .1.  `  X ) ) }  e.  _V )
308, 22, 7, 23, 29ovmpt2d 6201 1  |-  ( ph  ->  ( X S Y )  =  { <. 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 1652    e. wcel 1725   _Vcvv 2956   <.cop 3817   {copab 4265    X. cxp 4876   ` cfv 5454  (class class class)co 6081   Basecbs 13469    Hom chom 13540  compcco 13541   Catccat 13889   Idccid 13890  Sectcsect 13970
This theorem is referenced by:  sectss  13978  issect  13979
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-reu 2712  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-1st 6349  df-2nd 6350  df-sect 13973
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