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Theorem cidfn 13630
Description: The identity arrow operator is a function from objects to arrows. (Contributed by Mario Carneiro, 4-Jan-2017.)
Hypotheses
Ref Expression
cidfn.b  |-  B  =  ( Base `  C
)
cidfn.i  |-  .1.  =  ( Id `  C )
Assertion
Ref Expression
cidfn  |-  ( C  e.  Cat  ->  .1.  Fn  B )

Proof of Theorem cidfn
Dummy variables  f 
g  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 riotaex 6350 . . 3  |-  ( iota_ g  e.  ( x (  Hom  `  C )
x ) A. y  e.  B  ( A. f  e.  ( y
(  Hom  `  C ) x ) ( g ( <. y ,  x >. (comp `  C )
x ) f )  =  f  /\  A. f  e.  ( x
(  Hom  `  C ) y ) ( f ( <. x ,  x >. (comp `  C )
y ) g )  =  f ) )  e.  _V
2 eqid 2316 . . 3  |-  ( x  e.  B  |->  ( iota_ g  e.  ( x (  Hom  `  C )
x ) A. y  e.  B  ( A. f  e.  ( y
(  Hom  `  C ) x ) ( g ( <. y ,  x >. (comp `  C )
x ) f )  =  f  /\  A. f  e.  ( x
(  Hom  `  C ) y ) ( f ( <. x ,  x >. (comp `  C )
y ) g )  =  f ) ) )  =  ( x  e.  B  |->  ( iota_ g  e.  ( x (  Hom  `  C )
x ) A. y  e.  B  ( A. f  e.  ( y
(  Hom  `  C ) x ) ( g ( <. y ,  x >. (comp `  C )
x ) f )  =  f  /\  A. f  e.  ( x
(  Hom  `  C ) y ) ( f ( <. x ,  x >. (comp `  C )
y ) g )  =  f ) ) )
31, 2fnmpti 5409 . 2  |-  ( x  e.  B  |->  ( iota_ g  e.  ( x (  Hom  `  C )
x ) A. y  e.  B  ( A. f  e.  ( y
(  Hom  `  C ) x ) ( g ( <. y ,  x >. (comp `  C )
x ) f )  =  f  /\  A. f  e.  ( x
(  Hom  `  C ) y ) ( f ( <. x ,  x >. (comp `  C )
y ) g )  =  f ) ) )  Fn  B
4 cidfn.b . . . 4  |-  B  =  ( Base `  C
)
5 eqid 2316 . . . 4  |-  (  Hom  `  C )  =  (  Hom  `  C )
6 eqid 2316 . . . 4  |-  (comp `  C )  =  (comp `  C )
7 id 19 . . . 4  |-  ( C  e.  Cat  ->  C  e.  Cat )
8 cidfn.i . . . 4  |-  .1.  =  ( Id `  C )
94, 5, 6, 7, 8cidfval 13627 . . 3  |-  ( C  e.  Cat  ->  .1.  =  ( x  e.  B  |->  ( iota_ g  e.  ( x (  Hom  `  C ) x ) A. y  e.  B  ( A. f  e.  ( y (  Hom  `  C
) x ) ( g ( <. y ,  x >. (comp `  C
) x ) f )  =  f  /\  A. f  e.  ( x (  Hom  `  C
) y ) ( f ( <. x ,  x >. (comp `  C
) y ) g )  =  f ) ) ) )
109fneq1d 5372 . 2  |-  ( C  e.  Cat  ->  (  .1.  Fn  B  <->  ( x  e.  B  |->  ( iota_ g  e.  ( x (  Hom  `  C )
x ) A. y  e.  B  ( A. f  e.  ( y
(  Hom  `  C ) x ) ( g ( <. y ,  x >. (comp `  C )
x ) f )  =  f  /\  A. f  e.  ( x
(  Hom  `  C ) y ) ( f ( <. x ,  x >. (comp `  C )
y ) g )  =  f ) ) )  Fn  B ) )
113, 10mpbiri 224 1  |-  ( C  e.  Cat  ->  .1.  Fn  B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1633    e. wcel 1701   A.wral 2577   <.cop 3677    e. cmpt 4114    Fn wfn 5287   ` 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:  oppccatid  13671  fucidcl  13888  fucsect  13895  curfcl  14055  curf2ndf  14070
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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