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Theorem sectfval 13654
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 13653 . 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 732 . . . . . . 7  |-  ( (
ph  /\  ( x  =  X  /\  y  =  Y ) )  ->  x  =  X )
10 simprr 733 . . . . . . 7  |-  ( (
ph  /\  ( x  =  X  /\  y  =  Y ) )  -> 
y  =  Y )
119, 10oveq12d 5876 . . . . . 6  |-  ( (
ph  /\  ( x  =  X  /\  y  =  Y ) )  -> 
( x H y )  =  ( X H Y ) )
1211eleq2d 2350 . . . . 5  |-  ( (
ph  /\  ( x  =  X  /\  y  =  Y ) )  -> 
( f  e.  ( x H y )  <-> 
f  e.  ( X H Y ) ) )
1310, 9oveq12d 5876 . . . . . 6  |-  ( (
ph  /\  ( x  =  X  /\  y  =  Y ) )  -> 
( y H x )  =  ( Y H X ) )
1413eleq2d 2350 . . . . 5  |-  ( (
ph  /\  ( x  =  X  /\  y  =  Y ) )  -> 
( g  e.  ( y H x )  <-> 
g  e.  ( Y H X ) ) )
1512, 14anbi12d 691 . . . 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 3804 . . . . . . 7  |-  ( (
ph  /\  ( x  =  X  /\  y  =  Y ) )  ->  <. x ,  y >.  =  <. X ,  Y >. )
1716, 9oveq12d 5876 . . . . . 6  |-  ( (
ph  /\  ( x  =  X  /\  y  =  Y ) )  -> 
( <. x ,  y
>.  .x.  x )  =  ( <. X ,  Y >.  .x.  X ) )
1817oveqd 5875 . . . . 5  |-  ( (
ph  /\  ( x  =  X  /\  y  =  Y ) )  -> 
( g ( <.
x ,  y >.  .x.  x ) f )  =  ( g (
<. X ,  Y >.  .x. 
X ) f ) )
199fveq2d 5529 . . . . 5  |-  ( (
ph  /\  ( x  =  X  /\  y  =  Y ) )  -> 
(  .1.  `  x
)  =  (  .1.  `  X ) )
2018, 19eqeq12d 2297 . . . 4  |-  ( (
ph  /\  ( x  =  X  /\  y  =  Y ) )  -> 
( ( g (
<. x ,  y >.  .x.  x ) f )  =  (  .1.  `  x )  <->  ( g
( <. X ,  Y >.  .x.  X ) f )  =  (  .1.  `  X ) ) )
2115, 20anbi12d 691 . . 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 4082 . 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 5883 . . . . 5  |-  ( X H Y )  e. 
_V
25 ovex 5883 . . . . 5  |-  ( Y H X )  e. 
_V
2624, 25xpex 4801 . . . 4  |-  ( ( X H Y )  X.  ( Y H X ) )  e. 
_V
27 opabssxp 4762 . . . 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 4159 . . 3  |-  { <. f ,  g >.  |  ( ( f  e.  ( X H Y )  /\  g  e.  ( Y H X ) )  /\  ( g ( <. X ,  Y >.  .x.  X ) f )  =  (  .1.  `  X ) ) }  e.  _V
2928a1i 10 . 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 5975 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 358    = wceq 1623    e. wcel 1684   _Vcvv 2788   <.cop 3643   {copab 4076    X. cxp 4687   ` cfv 5255  (class class class)co 5858   Basecbs 13148    Hom chom 13219  compcco 13220   Catccat 13566   Idccid 13567  Sectcsect 13647
This theorem is referenced by:  sectss  13655  issect  13656
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-sect 13650
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