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Theorem homarcl 16001
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 3727 . 2  |-  ( F  e.  ( X H Y )  ->  -.  ( X H Y )  =  (/) )
2 homarcl.h . . . . 5  |-  H  =  (Homa
`  C )
3 df-homa 15999 . . . . . . . . 9  |- Homa  =  ( c  e.  Cat  |->  ( x  e.  ( ( Base `  c
)  X.  ( Base `  c ) )  |->  ( { x }  X.  ( ( Hom  `  c
) `  x )
) ) )
43dmmptss 5338 . . . . . . . 8  |-  dom Homa  C_  Cat
54sseli 3414 . . . . . . 7  |-  ( C  e.  dom Homa  ->  C  e.  Cat )
65con3i 142 . . . . . 6  |-  ( -.  C  e.  Cat  ->  -.  C  e.  dom Homa )
7 ndmfv 5903 . . . . . 6  |-  ( -.  C  e.  dom Homa  ->  (Homa
`  C )  =  (/) )
86, 7syl 17 . . . . 5  |-  ( -.  C  e.  Cat  ->  (Homa `  C )  =  (/) )
92, 8syl5eq 2517 . . . 4  |-  ( -.  C  e.  Cat  ->  H  =  (/) )
109oveqd 6325 . . 3  |-  ( -.  C  e.  Cat  ->  ( X H Y )  =  ( X (/) Y ) )
11 df-ov 6311 . . . 4  |-  ( X
(/) Y )  =  ( (/) `  <. X ,  Y >. )
12 0fv 5912 . . . 4  |-  ( (/) ` 
<. X ,  Y >. )  =  (/)
1311, 12eqtri 2493 . . 3  |-  ( X
(/) Y )  =  (/)
1410, 13syl6eq 2521 . 2  |-  ( -.  C  e.  Cat  ->  ( X H Y )  =  (/) )
151, 14nsyl2 132 1  |-  ( F  e.  ( X H Y )  ->  C  e.  Cat )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    = wceq 1452    e. wcel 1904   (/)c0 3722   {csn 3959   <.cop 3965    |-> cmpt 4454    X. cxp 4837   dom cdm 4839   ` cfv 5589  (class class class)co 6308   Basecbs 15199   Hom chom 15279   Catccat 15648  Homachoma 15996
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-rab 2765  df-v 3033  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-if 3873  df-sn 3960  df-pr 3962  df-op 3966  df-uni 4191  df-br 4396  df-opab 4455  df-mpt 4456  df-xp 4845  df-rel 4846  df-cnv 4847  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-iota 5553  df-fv 5597  df-ov 6311  df-homa 15999
This theorem is referenced by:  homarcl2  16008  homarel  16009  homa1  16010  homahom2  16011  coahom  16043  arwlid  16045  arwrid  16046  arwass  16047
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