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Theorem homarcl 14888
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 3637 . 2  |-  ( F  e.  ( X H Y )  ->  -.  ( X H Y )  =  (/) )
2 homarcl.h . . . . 5  |-  H  =  (Homa
`  C )
3 df-homa 14886 . . . . . . . . 9  |- Homa  =  ( c  e.  Cat  |->  ( x  e.  ( ( Base `  c
)  X.  ( Base `  c ) )  |->  ( { x }  X.  ( ( Hom  `  c
) `  x )
) ) )
43dmmptss 5329 . . . . . . . 8  |-  dom Homa  C_  Cat
54sseli 3347 . . . . . . 7  |-  ( C  e.  dom Homa  ->  C  e.  Cat )
65con3i 135 . . . . . 6  |-  ( -.  C  e.  Cat  ->  -.  C  e.  dom Homa )
7 ndmfv 5709 . . . . . 6  |-  ( -.  C  e.  dom Homa  ->  (Homa
`  C )  =  (/) )
86, 7syl 16 . . . . 5  |-  ( -.  C  e.  Cat  ->  (Homa `  C )  =  (/) )
92, 8syl5eq 2482 . . . 4  |-  ( -.  C  e.  Cat  ->  H  =  (/) )
109oveqd 6103 . . 3  |-  ( -.  C  e.  Cat  ->  ( X H Y )  =  ( X (/) Y ) )
11 df-ov 6089 . . . 4  |-  ( X
(/) Y )  =  ( (/) `  <. X ,  Y >. )
12 0fv 5718 . . . 4  |-  ( (/) ` 
<. X ,  Y >. )  =  (/)
1311, 12eqtri 2458 . . 3  |-  ( X
(/) Y )  =  (/)
1410, 13syl6eq 2486 . 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 1369    e. wcel 1756   (/)c0 3632   {csn 3872   <.cop 3878    e. cmpt 4345    X. cxp 4833   dom cdm 4835   ` cfv 5413  (class class class)co 6086   Basecbs 14166   Hom chom 14241   Catccat 14594  Homachoma 14883
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419  ax-sep 4408  ax-nul 4416  ax-pow 4465  ax-pr 4526
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2715  df-rex 2716  df-rab 2719  df-v 2969  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-nul 3633  df-if 3787  df-sn 3873  df-pr 3875  df-op 3879  df-uni 4087  df-br 4288  df-opab 4346  df-mpt 4347  df-xp 4841  df-rel 4842  df-cnv 4843  df-dm 4845  df-rn 4846  df-res 4847  df-ima 4848  df-iota 5376  df-fv 5421  df-ov 6089  df-homa 14886
This theorem is referenced by:  homarcl2  14895  homarel  14896  homa1  14897  homahom2  14898  coahom  14930  arwlid  14932  arwrid  14933  arwass  14934
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