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Theorem cidval 13579
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 13578 . 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 5876 . . 3  |-  ( (
ph  /\  x  =  X )  ->  (
x H x )  =  ( X H X ) )
9 eqidd 2284 . . . 4  |-  ( (
ph  /\  x  =  X )  ->  B  =  B )
107oveq2d 5874 . . . . . 6  |-  ( (
ph  /\  x  =  X )  ->  (
y H x )  =  ( y H X ) )
117opeq2d 3803 . . . . . . . . 9  |-  ( (
ph  /\  x  =  X )  ->  <. y ,  x >.  =  <. y ,  X >. )
1211, 7oveq12d 5876 . . . . . . . 8  |-  ( (
ph  /\  x  =  X )  ->  ( <. y ,  x >.  .x.  x )  =  (
<. y ,  X >.  .x. 
X ) )
1312oveqd 5875 . . . . . . 7  |-  ( (
ph  /\  x  =  X )  ->  (
g ( <. y ,  x >.  .x.  x ) f )  =  ( g ( <. y ,  X >.  .x.  X ) f ) )
1413eqeq1d 2291 . . . . . 6  |-  ( (
ph  /\  x  =  X )  ->  (
( g ( <.
y ,  x >.  .x.  x ) f )  =  f  <->  ( g
( <. y ,  X >.  .x.  X ) f )  =  f ) )
1510, 14raleqbidv 2748 . . . . 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 5873 . . . . . 6  |-  ( (
ph  /\  x  =  X )  ->  (
x H y )  =  ( X H y ) )
177, 7opeq12d 3804 . . . . . . . . 9  |-  ( (
ph  /\  x  =  X )  ->  <. x ,  x >.  =  <. X ,  X >. )
1817oveq1d 5873 . . . . . . . 8  |-  ( (
ph  /\  x  =  X )  ->  ( <. x ,  x >.  .x.  y )  =  (
<. X ,  X >.  .x.  y ) )
1918oveqd 5875 . . . . . . 7  |-  ( (
ph  /\  x  =  X )  ->  (
f ( <. x ,  x >.  .x.  y ) g )  =  ( f ( <. X ,  X >.  .x.  y )
g ) )
2019eqeq1d 2291 . . . . . 6  |-  ( (
ph  /\  x  =  X )  ->  (
( f ( <.
x ,  x >.  .x.  y ) g )  =  f  <->  ( f
( <. X ,  X >.  .x.  y ) g )  =  f ) )
2116, 20raleqbidv 2748 . . . . 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 2748 . . 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 6307 . 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 6308 . . 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 5606 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 1623    e. wcel 1684   A.wral 2543   _Vcvv 2788   <.cop 3643   ` cfv 5255  (class class class)co 5858   iota_crio 6297   Basecbs 13148    Hom chom 13219  compcco 13220   Catccat 13566   Idccid 13567
This theorem is referenced by:  catidcl  13584  catlid  13585  catrid  13586
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-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-pr 4214
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-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-riota 6304  df-cid 13571
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