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Theorem homarcl 15007
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 3743 . 2  |-  ( F  e.  ( X H Y )  ->  -.  ( X H Y )  =  (/) )
2 homarcl.h . . . . 5  |-  H  =  (Homa
`  C )
3 df-homa 15005 . . . . . . . . 9  |- Homa  =  ( c  e.  Cat  |->  ( x  e.  ( ( Base `  c
)  X.  ( Base `  c ) )  |->  ( { x }  X.  ( ( Hom  `  c
) `  x )
) ) )
43dmmptss 5435 . . . . . . . 8  |-  dom Homa  C_  Cat
54sseli 3453 . . . . . . 7  |-  ( C  e.  dom Homa  ->  C  e.  Cat )
65con3i 135 . . . . . 6  |-  ( -.  C  e.  Cat  ->  -.  C  e.  dom Homa )
7 ndmfv 5816 . . . . . 6  |-  ( -.  C  e.  dom Homa  ->  (Homa
`  C )  =  (/) )
86, 7syl 16 . . . . 5  |-  ( -.  C  e.  Cat  ->  (Homa `  C )  =  (/) )
92, 8syl5eq 2504 . . . 4  |-  ( -.  C  e.  Cat  ->  H  =  (/) )
109oveqd 6210 . . 3  |-  ( -.  C  e.  Cat  ->  ( X H Y )  =  ( X (/) Y ) )
11 df-ov 6196 . . . 4  |-  ( X
(/) Y )  =  ( (/) `  <. X ,  Y >. )
12 0fv 5825 . . . 4  |-  ( (/) ` 
<. X ,  Y >. )  =  (/)
1311, 12eqtri 2480 . . 3  |-  ( X
(/) Y )  =  (/)
1410, 13syl6eq 2508 . 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 1370    e. wcel 1758   (/)c0 3738   {csn 3978   <.cop 3984    |-> cmpt 4451    X. cxp 4939   dom cdm 4941   ` cfv 5519  (class class class)co 6193   Basecbs 14285   Hom chom 14360   Catccat 14713  Homachoma 15002
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-sep 4514  ax-nul 4522  ax-pow 4571  ax-pr 4632
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-rab 2804  df-v 3073  df-dif 3432  df-un 3434  df-in 3436  df-ss 3443  df-nul 3739  df-if 3893  df-sn 3979  df-pr 3981  df-op 3985  df-uni 4193  df-br 4394  df-opab 4452  df-mpt 4453  df-xp 4947  df-rel 4948  df-cnv 4949  df-dm 4951  df-rn 4952  df-res 4953  df-ima 4954  df-iota 5482  df-fv 5527  df-ov 6196  df-homa 15005
This theorem is referenced by:  homarcl2  15014  homarel  15015  homa1  15016  homahom2  15017  coahom  15049  arwlid  15051  arwrid  15052  arwass  15053
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