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Theorem catidcl 13600
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 2296 . . 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 13595 . 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 13593 . . 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 6335 . . 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 15 . 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 2370 1  |-  ( ph  ->  (  .1.  `  X
)  e.  ( X H X ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1632    e. wcel 1696   A.wral 2556   E!wreu 2558   <.cop 3656   ` cfv 5271  (class class class)co 5874   iota_crio 6313   Basecbs 13164    Hom chom 13235  compcco 13236   Catccat 13582   Idccid 13583
This theorem is referenced by:  oppccatid  13638  monsect  13697  fullsubc  13740  idfucl  13771  cofucl  13778  fthsect  13815  fucidcl  13855  idahom  13908  catcisolem  13954  xpccatid  13978  1stfcl  13987  2ndfcl  13988  prfcl  13993  evlfcl  14012  curf1cl  14018  curf2cl  14021  curfcl  14022  curfuncf  14028  uncfcurf  14029  diag12  14034  diag2  14035  curf2ndf  14037  hofcl  14049  yon12  14055  yon2  14056  yonedalem3a  14064  yonedalem3b  14069  yonedainv  14071
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-riota 6320  df-cat 13586  df-cid 13587
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