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Theorem cidval 13628
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 13627 . 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 447 . . . 4  |-  ( (
ph  /\  x  =  X )  ->  x  =  X )
87, 7oveq12d 5918 . . 3  |-  ( (
ph  /\  x  =  X )  ->  (
x H x )  =  ( X H X ) )
9 eqidd 2317 . . . 4  |-  ( (
ph  /\  x  =  X )  ->  B  =  B )
107oveq2d 5916 . . . . . 6  |-  ( (
ph  /\  x  =  X )  ->  (
y H x )  =  ( y H X ) )
117opeq2d 3840 . . . . . . . . 9  |-  ( (
ph  /\  x  =  X )  ->  <. y ,  x >.  =  <. y ,  X >. )
1211, 7oveq12d 5918 . . . . . . . 8  |-  ( (
ph  /\  x  =  X )  ->  ( <. y ,  x >.  .x.  x )  =  (
<. y ,  X >.  .x. 
X ) )
1312oveqd 5917 . . . . . . 7  |-  ( (
ph  /\  x  =  X )  ->  (
g ( <. y ,  x >.  .x.  x ) f )  =  ( g ( <. y ,  X >.  .x.  X ) f ) )
1413eqeq1d 2324 . . . . . 6  |-  ( (
ph  /\  x  =  X )  ->  (
( g ( <.
y ,  x >.  .x.  x ) f )  =  f  <->  ( g
( <. y ,  X >.  .x.  X ) f )  =  f ) )
1510, 14raleqbidv 2782 . . . . 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 ) )
167oveq1d 5915 . . . . . 6  |-  ( (
ph  /\  x  =  X )  ->  (
x H y )  =  ( X H y ) )
177, 7opeq12d 3841 . . . . . . . . 9  |-  ( (
ph  /\  x  =  X )  ->  <. x ,  x >.  =  <. X ,  X >. )
1817oveq1d 5915 . . . . . . . 8  |-  ( (
ph  /\  x  =  X )  ->  ( <. x ,  x >.  .x.  y )  =  (
<. X ,  X >.  .x.  y ) )
1918oveqd 5917 . . . . . . 7  |-  ( (
ph  /\  x  =  X )  ->  (
f ( <. x ,  x >.  .x.  y ) g )  =  ( f ( <. X ,  X >.  .x.  y )
g ) )
2019eqeq1d 2324 . . . . . 6  |-  ( (
ph  /\  x  =  X )  ->  (
( f ( <.
x ,  x >.  .x.  y ) g )  =  f  <->  ( f
( <. X ,  X >.  .x.  y ) g )  =  f ) )
2116, 20raleqbidv 2782 . . . . 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 ) )
2215, 21anbi12d 691 . . . 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 ) ) )
239, 22raleqbidv 2782 . . 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 ) ) )
248, 23riotaeqbidv 6349 . 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 ) ) )
25 cidval.x . 2  |-  ( ph  ->  X  e.  B )
26 riotaex 6350 . . 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
2726a1i 10 . 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 )
286, 24, 25, 27fvmptd 5644 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 358    = wceq 1633    e. wcel 1701   A.wral 2577   _Vcvv 2822   <.cop 3677   ` cfv 5292  (class class class)co 5900   iota_crio 6339   Basecbs 13195    Hom chom 13266  compcco 13267   Catccat 13615   Idccid 13616
This theorem is referenced by:  catidcl  13633  catlid  13634  catrid  13635
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-14 1705  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897  ax-ext 2297  ax-rep 4168  ax-sep 4178  ax-nul 4186  ax-pr 4251
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1533  df-nf 1536  df-sb 1640  df-eu 2180  df-mo 2181  df-clab 2303  df-cleq 2309  df-clel 2312  df-nfc 2441  df-ne 2481  df-ral 2582  df-rex 2583  df-reu 2584  df-rab 2586  df-v 2824  df-sbc 3026  df-csb 3116  df-dif 3189  df-un 3191  df-in 3193  df-ss 3200  df-nul 3490  df-if 3600  df-sn 3680  df-pr 3681  df-op 3683  df-uni 3865  df-iun 3944  df-br 4061  df-opab 4115  df-mpt 4116  df-id 4346  df-xp 4732  df-rel 4733  df-cnv 4734  df-co 4735  df-dm 4736  df-rn 4737  df-res 4738  df-ima 4739  df-iota 5256  df-fun 5294  df-fn 5295  df-f 5296  df-f1 5297  df-fo 5298  df-f1o 5299  df-fv 5300  df-ov 5903  df-riota 6346  df-cid 13620
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