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Theorem cidval 15084
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 15083 . 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 459 . . . 4  |-  ( (
ph  /\  x  =  X )  ->  x  =  X )
87, 7oveq12d 6214 . . 3  |-  ( (
ph  /\  x  =  X )  ->  (
x H x )  =  ( X H X ) )
97oveq2d 6212 . . . . . 6  |-  ( (
ph  /\  x  =  X )  ->  (
y H x )  =  ( y H X ) )
107opeq2d 4138 . . . . . . . . 9  |-  ( (
ph  /\  x  =  X )  ->  <. y ,  x >.  =  <. y ,  X >. )
1110, 7oveq12d 6214 . . . . . . . 8  |-  ( (
ph  /\  x  =  X )  ->  ( <. y ,  x >.  .x.  x )  =  (
<. y ,  X >.  .x. 
X ) )
1211oveqd 6213 . . . . . . 7  |-  ( (
ph  /\  x  =  X )  ->  (
g ( <. y ,  x >.  .x.  x ) f )  =  ( g ( <. y ,  X >.  .x.  X ) f ) )
1312eqeq1d 2384 . . . . . 6  |-  ( (
ph  /\  x  =  X )  ->  (
( g ( <.
y ,  x >.  .x.  x ) f )  =  f  <->  ( g
( <. y ,  X >.  .x.  X ) f )  =  f ) )
149, 13raleqbidv 2993 . . . . 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 6211 . . . . . 6  |-  ( (
ph  /\  x  =  X )  ->  (
x H y )  =  ( X H y ) )
167, 7opeq12d 4139 . . . . . . . . 9  |-  ( (
ph  /\  x  =  X )  ->  <. x ,  x >.  =  <. X ,  X >. )
1716oveq1d 6211 . . . . . . . 8  |-  ( (
ph  /\  x  =  X )  ->  ( <. x ,  x >.  .x.  y )  =  (
<. X ,  X >.  .x.  y ) )
1817oveqd 6213 . . . . . . 7  |-  ( (
ph  /\  x  =  X )  ->  (
f ( <. x ,  x >.  .x.  y ) g )  =  ( f ( <. X ,  X >.  .x.  y )
g ) )
1918eqeq1d 2384 . . . . . 6  |-  ( (
ph  /\  x  =  X )  ->  (
( f ( <.
x ,  x >.  .x.  y ) g )  =  f  <->  ( f
( <. X ,  X >.  .x.  y ) g )  =  f ) )
2015, 19raleqbidv 2993 . . . . 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 708 . . . 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 2821 . . 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 6161 . 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 6162 . . 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 5862 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 367    = wceq 1399    e. wcel 1826   A.wral 2732   _Vcvv 3034   <.cop 3950   ` cfv 5496   iota_crio 6157  (class class class)co 6196   Basecbs 14634   Hom chom 14713  compcco 14714   Catccat 15071   Idccid 15072
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-9 1830  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006  ax-ext 2360  ax-rep 4478  ax-sep 4488  ax-nul 4496  ax-pr 4601
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1402  df-ex 1621  df-nf 1625  df-sb 1748  df-eu 2222  df-mo 2223  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2532  df-ne 2579  df-ral 2737  df-rex 2738  df-reu 2739  df-rab 2741  df-v 3036  df-sbc 3253  df-csb 3349  df-dif 3392  df-un 3394  df-in 3396  df-ss 3403  df-nul 3712  df-if 3858  df-sn 3945  df-pr 3947  df-op 3951  df-uni 4164  df-iun 4245  df-br 4368  df-opab 4426  df-mpt 4427  df-id 4709  df-xp 4919  df-rel 4920  df-cnv 4921  df-co 4922  df-dm 4923  df-rn 4924  df-res 4925  df-ima 4926  df-iota 5460  df-fun 5498  df-fn 5499  df-f 5500  df-f1 5501  df-fo 5502  df-f1o 5503  df-fv 5504  df-riota 6158  df-ov 6199  df-cid 15076
This theorem is referenced by:  catidcl  15089  catlid  15090  catrid  15091
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