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Theorem catidcl 14616
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 2441 . . 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 14611 . 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 14609 . . 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 6065 . . 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 2515 1  |-  ( ph  ->  (  .1.  `  X
)  e.  ( X H X ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1364    e. wcel 1761   A.wral 2713   E!wreu 2715   <.cop 3880   ` cfv 5415   iota_crio 6048  (class class class)co 6090   Basecbs 14170   Hom chom 14245  compcco 14246   Catccat 14598   Idccid 14599
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-rep 4400  ax-sep 4410  ax-nul 4418  ax-pr 4528
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2261  df-mo 2262  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-ral 2718  df-rex 2719  df-reu 2720  df-rmo 2721  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-nul 3635  df-if 3789  df-sn 3875  df-pr 3877  df-op 3881  df-uni 4089  df-iun 4170  df-br 4290  df-opab 4348  df-mpt 4349  df-id 4632  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-riota 6049  df-ov 6093  df-cat 14602  df-cid 14603
This theorem is referenced by:  oppccatid  14654  monsect  14713  fullsubc  14756  idfucl  14787  cofucl  14794  fthsect  14831  fucidcl  14871  idahom  14924  catcisolem  14970  xpccatid  14994  1stfcl  15003  2ndfcl  15004  prfcl  15009  evlfcl  15028  curf1cl  15034  curf2cl  15037  curfcl  15038  curfuncf  15044  uncfcurf  15045  diag12  15050  diag2  15051  curf2ndf  15053  hofcl  15065  yon12  15071  yon2  15072  yonedalem3a  15080  yonedalem3b  15085  yonedainv  15087
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