MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  homarcl Unicode version

Theorem homarcl 13876
Description: Reverse closure for an arrow. (Contributed by Mario Carneiro, 11-Jan-2017.)
Hypothesis
Ref Expression
homarcl.h  |-  H  =  (Homa
`  C )
Assertion
Ref Expression
homarcl  |-  ( F  e.  ( X H Y )  ->  C  e.  Cat )

Proof of Theorem homarcl
Dummy variables  x  c are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 n0i 3473 . 2  |-  ( F  e.  ( X H Y )  ->  -.  ( X H Y )  =  (/) )
2 homarcl.h . . . . . 6  |-  H  =  (Homa
`  C )
3 df-homa 13874 . . . . . . . . . 10  |- Homa  =  ( c  e.  Cat  |->  ( x  e.  ( ( Base `  c
)  X.  ( Base `  c ) )  |->  ( { x }  X.  ( (  Hom  `  c
) `  x )
) ) )
43dmmptss 5185 . . . . . . . . 9  |-  dom Homa  C_  Cat
54sseli 3189 . . . . . . . 8  |-  ( C  e.  dom Homa  ->  C  e.  Cat )
65con3i 127 . . . . . . 7  |-  ( -.  C  e.  Cat  ->  -.  C  e.  dom Homa )
7 ndmfv 5568 . . . . . . 7  |-  ( -.  C  e.  dom Homa  ->  (Homa
`  C )  =  (/) )
86, 7syl 15 . . . . . 6  |-  ( -.  C  e.  Cat  ->  (Homa `  C )  =  (/) )
92, 8syl5eq 2340 . . . . 5  |-  ( -.  C  e.  Cat  ->  H  =  (/) )
109oveqd 5891 . . . 4  |-  ( -.  C  e.  Cat  ->  ( X H Y )  =  ( X (/) Y ) )
11 df-ov 5877 . . . . 5  |-  ( X
(/) Y )  =  ( (/) `  <. X ,  Y >. )
12 fv01 5575 . . . . 5  |-  ( (/) ` 
<. X ,  Y >. )  =  (/)
1311, 12eqtri 2316 . . . 4  |-  ( X
(/) Y )  =  (/)
1410, 13syl6eq 2344 . . 3  |-  ( -.  C  e.  Cat  ->  ( X H Y )  =  (/) )
1514con1i 121 . 2  |-  ( -.  ( X H Y )  =  (/)  ->  C  e.  Cat )
161, 15syl 15 1  |-  ( F  e.  ( X H Y )  ->  C  e.  Cat )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    = wceq 1632    e. wcel 1696   (/)c0 3468   {csn 3653   <.cop 3656    e. cmpt 4093    X. cxp 4703   dom cdm 4705   ` cfv 5271  (class class class)co 5874   Basecbs 13164    Hom chom 13235   Catccat 13582  Homachoma 13871
This theorem is referenced by:  homarcl2  13883  homarel  13884  homa1  13885  homahom2  13886  coahom  13918  arwlid  13920  arwrid  13921  arwass  13922
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-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  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-rab 2565  df-v 2803  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-br 4040  df-opab 4094  df-mpt 4095  df-xp 4711  df-rel 4712  df-cnv 4713  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fv 5279  df-ov 5877  df-homa 13874
  Copyright terms: Public domain W3C validator