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Theorem catidcl 15296
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 2402 . . 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 15291 . 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 15289 . . 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 6254 . . 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 17 . 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 2490 1  |-  ( ph  ->  (  .1.  `  X
)  e.  ( X H X ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    = wceq 1405    e. wcel 1842   A.wral 2754   E!wreu 2756   <.cop 3978   ` cfv 5569   iota_crio 6239  (class class class)co 6278   Basecbs 14841   Hom chom 14920  compcco 14921   Catccat 15278   Idccid 15279
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4507  ax-sep 4517  ax-nul 4525  ax-pr 4630
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2759  df-rex 2760  df-reu 2761  df-rmo 2762  df-rab 2763  df-v 3061  df-sbc 3278  df-csb 3374  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-nul 3739  df-if 3886  df-sn 3973  df-pr 3975  df-op 3979  df-uni 4192  df-iun 4273  df-br 4396  df-opab 4454  df-mpt 4455  df-id 4738  df-xp 4829  df-rel 4830  df-cnv 4831  df-co 4832  df-dm 4833  df-rn 4834  df-res 4835  df-ima 4836  df-iota 5533  df-fun 5571  df-fn 5572  df-f 5573  df-f1 5574  df-fo 5575  df-f1o 5576  df-fv 5577  df-riota 6240  df-ov 6281  df-cat 15282  df-cid 15283
This theorem is referenced by:  oppccatid  15332  monsect  15396  sectid  15399  cicref  15414  catsubcat  15452  fullsubc  15463  idfucl  15494  cofucl  15501  fthsect  15538  fucidcl  15578  initoid  15608  termoid  15609  idahom  15663  catcisolem  15709  xpccatid  15781  1stfcl  15790  2ndfcl  15791  prfcl  15796  evlfcl  15815  curf1cl  15821  curf2cl  15824  curfcl  15825  curfuncf  15831  uncfcurf  15832  diag12  15837  diag2  15838  curf2ndf  15840  hofcl  15852  yon12  15858  yon2  15859  yonedalem3a  15867  yonedalem3b  15872  yonedainv  15874
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