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Theorem cidval 13829
Description: Each object in a category has an associated identity arrow. (Contributed by Mario Carneiro, 2-Jan-2017.)
Hypotheses
Ref Expression
cidfval.b  |-  B  =  ( Base `  C
)
cidfval.h  |-  H  =  (  Hom  `  C
)
cidfval.o  |-  .x.  =  (comp `  C )
cidfval.c  |-  ( ph  ->  C  e.  Cat )
cidfval.i  |-  .1.  =  ( Id `  C )
cidval.x  |-  ( ph  ->  X  e.  B )
Assertion
Ref Expression
cidval  |-  ( ph  ->  (  .1.  `  X
)  =  ( iota_ g  e.  ( X H X ) A. y  e.  B  ( A. f  e.  ( y H X ) ( g ( <. y ,  X >.  .x.  X ) f )  =  f  /\  A. f  e.  ( X H y ) ( f ( <. X ,  X >.  .x.  y )
g )  =  f ) ) )
Distinct variable groups:    f, g,
y, B    C, f,
g, y    .x. , f, g, y    f, H, g, y    ph, f, g, y   
f, X, g, y
Allowed substitution hints:    .1. ( y, f, g)

Proof of Theorem cidval
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 cidfval.b . . 3  |-  B  =  ( Base `  C
)
2 cidfval.h . . 3  |-  H  =  (  Hom  `  C
)
3 cidfval.o . . 3  |-  .x.  =  (comp `  C )
4 cidfval.c . . 3  |-  ( ph  ->  C  e.  Cat )
5 cidfval.i . . 3  |-  .1.  =  ( Id `  C )
61, 2, 3, 4, 5cidfval 13828 . 2  |-  ( ph  ->  .1.  =  ( x  e.  B  |->  ( iota_ g  e.  ( x H x ) A. y  e.  B  ( A. f  e.  ( y H x ) ( g ( <. y ,  x >.  .x.  x ) f )  =  f  /\  A. f  e.  ( x H y ) ( f (
<. x ,  x >.  .x.  y ) g )  =  f ) ) ) )
7 simpr 448 . . . 4  |-  ( (
ph  /\  x  =  X )  ->  x  =  X )
87, 7oveq12d 6038 . . 3  |-  ( (
ph  /\  x  =  X )  ->  (
x H x )  =  ( X H X ) )
97oveq2d 6036 . . . . . 6  |-  ( (
ph  /\  x  =  X )  ->  (
y H x )  =  ( y H X ) )
107opeq2d 3933 . . . . . . . . 9  |-  ( (
ph  /\  x  =  X )  ->  <. y ,  x >.  =  <. y ,  X >. )
1110, 7oveq12d 6038 . . . . . . . 8  |-  ( (
ph  /\  x  =  X )  ->  ( <. y ,  x >.  .x.  x )  =  (
<. y ,  X >.  .x. 
X ) )
1211oveqd 6037 . . . . . . 7  |-  ( (
ph  /\  x  =  X )  ->  (
g ( <. y ,  x >.  .x.  x ) f )  =  ( g ( <. y ,  X >.  .x.  X ) f ) )
1312eqeq1d 2395 . . . . . 6  |-  ( (
ph  /\  x  =  X )  ->  (
( g ( <.
y ,  x >.  .x.  x ) f )  =  f  <->  ( g
( <. y ,  X >.  .x.  X ) f )  =  f ) )
149, 13raleqbidv 2859 . . . . 5  |-  ( (
ph  /\  x  =  X )  ->  ( A. f  e.  (
y H x ) ( g ( <.
y ,  x >.  .x.  x ) f )  =  f  <->  A. f  e.  ( y H X ) ( g (
<. y ,  X >.  .x. 
X ) f )  =  f ) )
157oveq1d 6035 . . . . . 6  |-  ( (
ph  /\  x  =  X )  ->  (
x H y )  =  ( X H y ) )
167, 7opeq12d 3934 . . . . . . . . 9  |-  ( (
ph  /\  x  =  X )  ->  <. x ,  x >.  =  <. X ,  X >. )
1716oveq1d 6035 . . . . . . . 8  |-  ( (
ph  /\  x  =  X )  ->  ( <. x ,  x >.  .x.  y )  =  (
<. X ,  X >.  .x.  y ) )
1817oveqd 6037 . . . . . . 7  |-  ( (
ph  /\  x  =  X )  ->  (
f ( <. x ,  x >.  .x.  y ) g )  =  ( f ( <. X ,  X >.  .x.  y )
g ) )
1918eqeq1d 2395 . . . . . 6  |-  ( (
ph  /\  x  =  X )  ->  (
( f ( <.
x ,  x >.  .x.  y ) g )  =  f  <->  ( f
( <. X ,  X >.  .x.  y ) g )  =  f ) )
2015, 19raleqbidv 2859 . . . . 5  |-  ( (
ph  /\  x  =  X )  ->  ( A. f  e.  (
x H y ) ( f ( <.
x ,  x >.  .x.  y ) g )  =  f  <->  A. f  e.  ( X H y ) ( f (
<. X ,  X >.  .x.  y ) g )  =  f ) )
2114, 20anbi12d 692 . . . 4  |-  ( (
ph  /\  x  =  X )  ->  (
( A. f  e.  ( y H x ) ( g (
<. y ,  x >.  .x.  x ) f )  =  f  /\  A. f  e.  ( x H y ) ( f ( <. x ,  x >.  .x.  y ) g )  =  f )  <->  ( A. f  e.  ( y H X ) ( g (
<. y ,  X >.  .x. 
X ) f )  =  f  /\  A. f  e.  ( X H y ) ( f ( <. X ,  X >.  .x.  y )
g )  =  f ) ) )
2221ralbidv 2669 . . 3  |-  ( (
ph  /\  x  =  X )  ->  ( A. y  e.  B  ( A. f  e.  ( y H x ) ( g ( <.
y ,  x >.  .x.  x ) f )  =  f  /\  A. f  e.  ( x H y ) ( f ( <. x ,  x >.  .x.  y ) g )  =  f )  <->  A. y  e.  B  ( A. f  e.  ( y H X ) ( g ( <.
y ,  X >.  .x. 
X ) f )  =  f  /\  A. f  e.  ( X H y ) ( f ( <. X ,  X >.  .x.  y )
g )  =  f ) ) )
238, 22riotaeqbidv 6488 . 2  |-  ( (
ph  /\  x  =  X )  ->  ( iota_ g  e.  ( x H x ) A. y  e.  B  ( A. f  e.  (
y H x ) ( g ( <.
y ,  x >.  .x.  x ) f )  =  f  /\  A. f  e.  ( x H y ) ( f ( <. x ,  x >.  .x.  y ) g )  =  f ) )  =  (
iota_ g  e.  ( X H X ) A. y  e.  B  ( A. f  e.  (
y H X ) ( g ( <.
y ,  X >.  .x. 
X ) f )  =  f  /\  A. f  e.  ( X H y ) ( f ( <. X ,  X >.  .x.  y )
g )  =  f ) ) )
24 cidval.x . 2  |-  ( ph  ->  X  e.  B )
25 riotaex 6489 . . 3  |-  ( iota_ g  e.  ( X H X ) A. y  e.  B  ( A. f  e.  ( y H X ) ( g ( <. y ,  X >.  .x.  X ) f )  =  f  /\  A. f  e.  ( X H y ) ( f ( <. X ,  X >.  .x.  y )
g )  =  f ) )  e.  _V
2625a1i 11 . 2  |-  ( ph  ->  ( iota_ g  e.  ( X H X ) A. y  e.  B  ( A. f  e.  ( y H X ) ( g ( <.
y ,  X >.  .x. 
X ) f )  =  f  /\  A. f  e.  ( X H y ) ( f ( <. X ,  X >.  .x.  y )
g )  =  f ) )  e.  _V )
276, 23, 24, 26fvmptd 5749 1  |-  ( ph  ->  (  .1.  `  X
)  =  ( iota_ g  e.  ( X H X ) A. y  e.  B  ( A. f  e.  ( y H X ) ( g ( <. y ,  X >.  .x.  X ) f )  =  f  /\  A. f  e.  ( X H y ) ( f ( <. X ,  X >.  .x.  y )
g )  =  f ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1717   A.wral 2649   _Vcvv 2899   <.cop 3760   ` cfv 5394  (class class class)co 6020   iota_crio 6478   Basecbs 13396    Hom chom 13467  compcco 13468   Catccat 13816   Idccid 13817
This theorem is referenced by:  catidcl  13834  catlid  13835  catrid  13836
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-rep 4261  ax-sep 4271  ax-nul 4279  ax-pr 4344
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-ral 2654  df-rex 2655  df-reu 2656  df-rab 2658  df-v 2901  df-sbc 3105  df-csb 3195  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-nul 3572  df-if 3683  df-sn 3763  df-pr 3764  df-op 3766  df-uni 3958  df-iun 4037  df-br 4154  df-opab 4208  df-mpt 4209  df-id 4439  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-rn 4829  df-res 4830  df-ima 4831  df-iota 5358  df-fun 5396  df-fn 5397  df-f 5398  df-f1 5399  df-fo 5400  df-f1o 5401  df-fv 5402  df-ov 6023  df-riota 6485  df-cid 13821
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