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Theorem cidval 14949
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 14948 . 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 461 . . . 4  |-  ( (
ph  /\  x  =  X )  ->  x  =  X )
87, 7oveq12d 6313 . . 3  |-  ( (
ph  /\  x  =  X )  ->  (
x H x )  =  ( X H X ) )
97oveq2d 6311 . . . . . 6  |-  ( (
ph  /\  x  =  X )  ->  (
y H x )  =  ( y H X ) )
107opeq2d 4226 . . . . . . . . 9  |-  ( (
ph  /\  x  =  X )  ->  <. y ,  x >.  =  <. y ,  X >. )
1110, 7oveq12d 6313 . . . . . . . 8  |-  ( (
ph  /\  x  =  X )  ->  ( <. y ,  x >.  .x.  x )  =  (
<. y ,  X >.  .x. 
X ) )
1211oveqd 6312 . . . . . . 7  |-  ( (
ph  /\  x  =  X )  ->  (
g ( <. y ,  x >.  .x.  x ) f )  =  ( g ( <. y ,  X >.  .x.  X ) f ) )
1312eqeq1d 2469 . . . . . 6  |-  ( (
ph  /\  x  =  X )  ->  (
( g ( <.
y ,  x >.  .x.  x ) f )  =  f  <->  ( g
( <. y ,  X >.  .x.  X ) f )  =  f ) )
149, 13raleqbidv 3077 . . . . 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 6310 . . . . . 6  |-  ( (
ph  /\  x  =  X )  ->  (
x H y )  =  ( X H y ) )
167, 7opeq12d 4227 . . . . . . . . 9  |-  ( (
ph  /\  x  =  X )  ->  <. x ,  x >.  =  <. X ,  X >. )
1716oveq1d 6310 . . . . . . . 8  |-  ( (
ph  /\  x  =  X )  ->  ( <. x ,  x >.  .x.  y )  =  (
<. X ,  X >.  .x.  y ) )
1817oveqd 6312 . . . . . . 7  |-  ( (
ph  /\  x  =  X )  ->  (
f ( <. x ,  x >.  .x.  y ) g )  =  ( f ( <. X ,  X >.  .x.  y )
g ) )
1918eqeq1d 2469 . . . . . 6  |-  ( (
ph  /\  x  =  X )  ->  (
( f ( <.
x ,  x >.  .x.  y ) g )  =  f  <->  ( f
( <. X ,  X >.  .x.  y ) g )  =  f ) )
2015, 19raleqbidv 3077 . . . . 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 710 . . . 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 2906 . . 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 6259 . 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 6260 . . 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 5962 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 setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1379    e. wcel 1767   A.wral 2817   _Vcvv 3118   <.cop 4039   ` cfv 5594   iota_crio 6255  (class class class)co 6295   Basecbs 14507   Hom chom 14583  compcco 14584   Catccat 14936   Idccid 14937
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4564  ax-sep 4574  ax-nul 4582  ax-pr 4692
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-reu 2824  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-sn 4034  df-pr 4036  df-op 4040  df-uni 4252  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-id 4801  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6256  df-ov 6298  df-cid 14941
This theorem is referenced by:  catidcl  14954  catlid  14955  catrid  14956
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