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Theorem catidcl 14635
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 2443 . . 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 14630 . 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 14628 . . 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 6082 . . 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 2517 1  |-  ( ph  ->  (  .1.  `  X
)  e.  ( X H X ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1369    e. wcel 1756   A.wral 2730   E!wreu 2732   <.cop 3898   ` cfv 5433   iota_crio 6066  (class class class)co 6106   Basecbs 14189   Hom chom 14264  compcco 14265   Catccat 14617   Idccid 14618
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4418  ax-sep 4428  ax-nul 4436  ax-pr 4546
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2622  df-ral 2735  df-rex 2736  df-reu 2737  df-rmo 2738  df-rab 2739  df-v 2989  df-sbc 3202  df-csb 3304  df-dif 3346  df-un 3348  df-in 3350  df-ss 3357  df-nul 3653  df-if 3807  df-sn 3893  df-pr 3895  df-op 3899  df-uni 4107  df-iun 4188  df-br 4308  df-opab 4366  df-mpt 4367  df-id 4651  df-xp 4861  df-rel 4862  df-cnv 4863  df-co 4864  df-dm 4865  df-rn 4866  df-res 4867  df-ima 4868  df-iota 5396  df-fun 5435  df-fn 5436  df-f 5437  df-f1 5438  df-fo 5439  df-f1o 5440  df-fv 5441  df-riota 6067  df-ov 6109  df-cat 14621  df-cid 14622
This theorem is referenced by:  oppccatid  14673  monsect  14732  fullsubc  14775  idfucl  14806  cofucl  14813  fthsect  14850  fucidcl  14890  idahom  14943  catcisolem  14989  xpccatid  15013  1stfcl  15022  2ndfcl  15023  prfcl  15028  evlfcl  15047  curf1cl  15053  curf2cl  15056  curfcl  15057  curfuncf  15063  uncfcurf  15064  diag12  15069  diag2  15070  curf2ndf  15072  hofcl  15084  yon12  15090  yon2  15091  yonedalem3a  15099  yonedalem3b  15104  yonedainv  15106
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