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Theorem catidcl 13909
Description: Each object in a category has an associated identity arrow. (Contributed by Mario Carneiro, 2-Jan-2017.)
Hypotheses
Ref Expression
catidcl.b  |-  B  =  ( Base `  C
)
catidcl.h  |-  H  =  (  Hom  `  C
)
catidcl.i  |-  .1.  =  ( Id `  C )
catidcl.c  |-  ( ph  ->  C  e.  Cat )
catidcl.x  |-  ( ph  ->  X  e.  B )
Assertion
Ref Expression
catidcl  |-  ( ph  ->  (  .1.  `  X
)  e.  ( X H X ) )

Proof of Theorem catidcl
Dummy variables  f 
g  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 catidcl.b . . 3  |-  B  =  ( Base `  C
)
2 catidcl.h . . 3  |-  H  =  (  Hom  `  C
)
3 eqid 2438 . . 3  |-  (comp `  C )  =  (comp `  C )
4 catidcl.c . . 3  |-  ( ph  ->  C  e.  Cat )
5 catidcl.i . . 3  |-  .1.  =  ( Id `  C )
6 catidcl.x . . 3  |-  ( ph  ->  X  e.  B )
71, 2, 3, 4, 5, 6cidval 13904 . 2  |-  ( ph  ->  (  .1.  `  X
)  =  ( iota_ g  e.  ( X H X ) A. y  e.  B  ( A. f  e.  ( y H X ) ( g ( <. y ,  X >. (comp `  C ) X ) f )  =  f  /\  A. f  e.  ( X H y ) ( f ( <. X ,  X >. (comp `  C
) y ) g )  =  f ) ) )
81, 2, 3, 4, 6catideu 13902 . . 3  |-  ( ph  ->  E! g  e.  ( X H X ) A. y  e.  B  ( A. f  e.  ( y H X ) ( g ( <.
y ,  X >. (comp `  C ) X ) f )  =  f  /\  A. f  e.  ( X H y ) ( f (
<. X ,  X >. (comp `  C ) y ) g )  =  f ) )
9 riotacl 6566 . . 3  |-  ( E! g  e.  ( X H X ) A. y  e.  B  ( A. f  e.  (
y H X ) ( g ( <.
y ,  X >. (comp `  C ) X ) f )  =  f  /\  A. f  e.  ( X H y ) ( f (
<. X ,  X >. (comp `  C ) y ) g )  =  f )  ->  ( iota_ g  e.  ( X H X ) A. y  e.  B  ( A. f  e.  ( y H X ) ( g ( <. y ,  X >. (comp `  C ) X ) f )  =  f  /\  A. f  e.  ( X H y ) ( f ( <. X ,  X >. (comp `  C
) y ) g )  =  f ) )  e.  ( X H X ) )
108, 9syl 16 . 2  |-  ( ph  ->  ( iota_ g  e.  ( X H X ) A. y  e.  B  ( A. f  e.  ( y H X ) ( g ( <.
y ,  X >. (comp `  C ) X ) f )  =  f  /\  A. f  e.  ( X H y ) ( f (
<. X ,  X >. (comp `  C ) y ) g )  =  f ) )  e.  ( X H X ) )
117, 10eqeltrd 2512 1  |-  ( ph  ->  (  .1.  `  X
)  e.  ( X H X ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360    = wceq 1653    e. wcel 1726   A.wral 2707   E!wreu 2709   <.cop 3819   ` cfv 5456  (class class class)co 6083   iota_crio 6544   Basecbs 13471    Hom chom 13542  compcco 13543   Catccat 13891   Idccid 13892
This theorem is referenced by:  oppccatid  13947  monsect  14006  fullsubc  14049  idfucl  14080  cofucl  14087  fthsect  14124  fucidcl  14164  idahom  14217  catcisolem  14263  xpccatid  14287  1stfcl  14296  2ndfcl  14297  prfcl  14302  evlfcl  14321  curf1cl  14327  curf2cl  14330  curfcl  14331  curfuncf  14337  uncfcurf  14338  diag12  14343  diag2  14344  curf2ndf  14346  hofcl  14358  yon12  14364  yon2  14365  yonedalem3a  14373  yonedalem3b  14378  yonedainv  14380
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4322  ax-sep 4332  ax-nul 4340  ax-pr 4405
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-reu 2714  df-rmo 2715  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-iun 4097  df-br 4215  df-opab 4269  df-mpt 4270  df-id 4500  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-ov 6086  df-riota 6551  df-cat 13895  df-cid 13896
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