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Theorem homarcl 14185
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 3635 . 2  |-  ( F  e.  ( X H Y )  ->  -.  ( X H Y )  =  (/) )
2 homarcl.h . . . . 5  |-  H  =  (Homa
`  C )
3 df-homa 14183 . . . . . . . . 9  |- Homa  =  ( c  e.  Cat  |->  ( x  e.  ( ( Base `  c
)  X.  ( Base `  c ) )  |->  ( { x }  X.  ( (  Hom  `  c
) `  x )
) ) )
43dmmptss 5368 . . . . . . . 8  |-  dom Homa  C_  Cat
54sseli 3346 . . . . . . 7  |-  ( C  e.  dom Homa  ->  C  e.  Cat )
65con3i 130 . . . . . 6  |-  ( -.  C  e.  Cat  ->  -.  C  e.  dom Homa )
7 ndmfv 5757 . . . . . 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 6100 . . 3  |-  ( -.  C  e.  Cat  ->  ( X H Y )  =  ( X (/) Y ) )
11 df-ov 6086 . . . 4  |-  ( X
(/) Y )  =  ( (/) `  <. X ,  Y >. )
12 fv01 5765 . . . 4  |-  ( (/) ` 
<. X ,  Y >. )  =  (/)
1311, 12eqtri 2458 . . 3  |-  ( X
(/) Y )  =  (/)
1410, 13syl6eq 2486 . 2  |-  ( -.  C  e.  Cat  ->  ( X H Y )  =  (/) )
151, 14nsyl2 122 1  |-  ( F  e.  ( X H Y )  ->  C  e.  Cat )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    = wceq 1653    e. wcel 1726   (/)c0 3630   {csn 3816   <.cop 3819    e. cmpt 4268    X. cxp 4878   dom cdm 4880   ` cfv 5456  (class class class)co 6083   Basecbs 13471    Hom chom 13542   Catccat 13891  Homachoma 14180
This theorem is referenced by:  homarcl2  14192  homarel  14193  homa1  14194  homahom2  14195  coahom  14227  arwlid  14229  arwrid  14230  arwass  14231
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4215  df-opab 4269  df-mpt 4270  df-xp 4886  df-rel 4887  df-cnv 4888  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fv 5464  df-ov 6086  df-homa 14183
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