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Theorem homarcl 13860
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 3460 . 2  |-  ( F  e.  ( X H Y )  ->  -.  ( X H Y )  =  (/) )
2 homarcl.h . . . . . 6  |-  H  =  (Homa
`  C )
3 df-homa 13858 . . . . . . . . . 10  |- Homa  =  ( c  e.  Cat  |->  ( x  e.  ( ( Base `  c
)  X.  ( Base `  c ) )  |->  ( { x }  X.  ( (  Hom  `  c
) `  x )
) ) )
43dmmptss 5169 . . . . . . . . 9  |-  dom Homa  C_  Cat
54sseli 3176 . . . . . . . 8  |-  ( C  e.  dom Homa  ->  C  e.  Cat )
65con3i 127 . . . . . . 7  |-  ( -.  C  e.  Cat  ->  -.  C  e.  dom Homa )
7 ndmfv 5552 . . . . . . 7  |-  ( -.  C  e.  dom Homa  ->  (Homa
`  C )  =  (/) )
86, 7syl 15 . . . . . 6  |-  ( -.  C  e.  Cat  ->  (Homa `  C )  =  (/) )
92, 8syl5eq 2327 . . . . 5  |-  ( -.  C  e.  Cat  ->  H  =  (/) )
109oveqd 5875 . . . 4  |-  ( -.  C  e.  Cat  ->  ( X H Y )  =  ( X (/) Y ) )
11 df-ov 5861 . . . . 5  |-  ( X
(/) Y )  =  ( (/) `  <. X ,  Y >. )
12 fv01 5559 . . . . 5  |-  ( (/) ` 
<. X ,  Y >. )  =  (/)
1311, 12eqtri 2303 . . . 4  |-  ( X
(/) Y )  =  (/)
1410, 13syl6eq 2331 . . 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 1623    e. wcel 1684   (/)c0 3455   {csn 3640   <.cop 3643    e. cmpt 4077    X. cxp 4687   dom cdm 4689   ` cfv 5255  (class class class)co 5858   Basecbs 13148    Hom chom 13219   Catccat 13566  Homachoma 13855
This theorem is referenced by:  homarcl2  13867  homarel  13868  homa1  13869  homahom2  13870  coahom  13902  arwlid  13904  arwrid  13905  arwass  13906
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-xp 4695  df-rel 4696  df-cnv 4697  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fv 5263  df-ov 5861  df-homa 13858
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