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

Theorem homarcl 15629
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 3742 . 2  |-  ( F  e.  ( X H Y )  ->  -.  ( X H Y )  =  (/) )
2 homarcl.h . . . . 5  |-  H  =  (Homa
`  C )
3 df-homa 15627 . . . . . . . . 9  |- Homa  =  ( c  e.  Cat  |->  ( x  e.  ( ( Base `  c
)  X.  ( Base `  c ) )  |->  ( { x }  X.  ( ( Hom  `  c
) `  x )
) ) )
43dmmptss 5318 . . . . . . . 8  |-  dom Homa  C_  Cat
54sseli 3437 . . . . . . 7  |-  ( C  e.  dom Homa  ->  C  e.  Cat )
65con3i 135 . . . . . 6  |-  ( -.  C  e.  Cat  ->  -.  C  e.  dom Homa )
7 ndmfv 5872 . . . . . 6  |-  ( -.  C  e.  dom Homa  ->  (Homa
`  C )  =  (/) )
86, 7syl 17 . . . . 5  |-  ( -.  C  e.  Cat  ->  (Homa `  C )  =  (/) )
92, 8syl5eq 2455 . . . 4  |-  ( -.  C  e.  Cat  ->  H  =  (/) )
109oveqd 6294 . . 3  |-  ( -.  C  e.  Cat  ->  ( X H Y )  =  ( X (/) Y ) )
11 df-ov 6280 . . . 4  |-  ( X
(/) Y )  =  ( (/) `  <. X ,  Y >. )
12 0fv 5881 . . . 4  |-  ( (/) ` 
<. X ,  Y >. )  =  (/)
1311, 12eqtri 2431 . . 3  |-  ( X
(/) Y )  =  (/)
1410, 13syl6eq 2459 . 2  |-  ( -.  C  e.  Cat  ->  ( X H Y )  =  (/) )
151, 14nsyl2 127 1  |-  ( F  e.  ( X H Y )  ->  C  e.  Cat )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    = wceq 1405    e. wcel 1842   (/)c0 3737   {csn 3971   <.cop 3977    |-> cmpt 4452    X. cxp 4820   dom cdm 4822   ` cfv 5568  (class class class)co 6277   Basecbs 14839   Hom chom 14918   Catccat 15276  Homachoma 15624
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-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-sep 4516  ax-nul 4524  ax-pow 4571  ax-pr 4629
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 2758  df-rex 2759  df-rab 2762  df-v 3060  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-nul 3738  df-if 3885  df-sn 3972  df-pr 3974  df-op 3978  df-uni 4191  df-br 4395  df-opab 4453  df-mpt 4454  df-xp 4828  df-rel 4829  df-cnv 4830  df-dm 4832  df-rn 4833  df-res 4834  df-ima 4835  df-iota 5532  df-fv 5576  df-ov 6280  df-homa 15627
This theorem is referenced by:  homarcl2  15636  homarel  15637  homa1  15638  homahom2  15639  coahom  15671  arwlid  15673  arwrid  15674  arwass  15675
  Copyright terms: Public domain W3C validator