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Theorem arwval 16016
Description: The set of arrows is the union of all the disjointified hom-sets. (Contributed by Mario Carneiro, 11-Jan-2017.)
Hypotheses
Ref Expression
arwval.a  |-  A  =  (Nat `  C )
arwval.h  |-  H  =  (Homa
`  C )
Assertion
Ref Expression
arwval  |-  A  = 
U. ran  H

Proof of Theorem arwval
Dummy variables  x  c are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 arwval.a . 2  |-  A  =  (Nat `  C )
2 fveq2 5879 . . . . . . 7  |-  ( c  =  C  ->  (Homa `  c
)  =  (Homa `  C
) )
3 arwval.h . . . . . . 7  |-  H  =  (Homa
`  C )
42, 3syl6eqr 2523 . . . . . 6  |-  ( c  =  C  ->  (Homa `  c
)  =  H )
54rneqd 5068 . . . . 5  |-  ( c  =  C  ->  ran  (Homa `  c )  =  ran  H )
65unieqd 4200 . . . 4  |-  ( c  =  C  ->  U. ran  (Homa `  c )  =  U. ran  H )
7 df-arw 16000 . . . 4  |- Nat  =  ( c  e.  Cat  |->  U.
ran  (Homa
`  c ) )
8 fvex 5889 . . . . . . 7  |-  (Homa `  C
)  e.  _V
93, 8eqeltri 2545 . . . . . 6  |-  H  e. 
_V
109rnex 6746 . . . . 5  |-  ran  H  e.  _V
1110uniex 6606 . . . 4  |-  U. ran  H  e.  _V
126, 7, 11fvmpt 5963 . . 3  |-  ( C  e.  Cat  ->  (Nat `  C )  =  U. ran  H )
137dmmptss 5338 . . . . . . 7  |-  dom Nat  C_  Cat
1413sseli 3414 . . . . . 6  |-  ( C  e.  dom Nat  ->  C  e. 
Cat )
1514con3i 142 . . . . 5  |-  ( -.  C  e.  Cat  ->  -.  C  e.  dom Nat )
16 ndmfv 5903 . . . . 5  |-  ( -.  C  e.  dom Nat  ->  (Nat
`  C )  =  (/) )
1715, 16syl 17 . . . 4  |-  ( -.  C  e.  Cat  ->  (Nat
`  C )  =  (/) )
18 df-homa 15999 . . . . . . . . . . . . 13  |- Homa  =  ( c  e.  Cat  |->  ( x  e.  ( ( Base `  c
)  X.  ( Base `  c ) )  |->  ( { x }  X.  ( ( Hom  `  c
) `  x )
) ) )
1918dmmptss 5338 . . . . . . . . . . . 12  |-  dom Homa  C_  Cat
2019sseli 3414 . . . . . . . . . . 11  |-  ( C  e.  dom Homa  ->  C  e.  Cat )
2120con3i 142 . . . . . . . . . 10  |-  ( -.  C  e.  Cat  ->  -.  C  e.  dom Homa )
22 ndmfv 5903 . . . . . . . . . 10  |-  ( -.  C  e.  dom Homa  ->  (Homa
`  C )  =  (/) )
2321, 22syl 17 . . . . . . . . 9  |-  ( -.  C  e.  Cat  ->  (Homa `  C )  =  (/) )
243, 23syl5eq 2517 . . . . . . . 8  |-  ( -.  C  e.  Cat  ->  H  =  (/) )
2524rneqd 5068 . . . . . . 7  |-  ( -.  C  e.  Cat  ->  ran 
H  =  ran  (/) )
26 rn0 5092 . . . . . . 7  |-  ran  (/)  =  (/)
2725, 26syl6eq 2521 . . . . . 6  |-  ( -.  C  e.  Cat  ->  ran 
H  =  (/) )
2827unieqd 4200 . . . . 5  |-  ( -.  C  e.  Cat  ->  U.
ran  H  =  U. (/) )
29 uni0 4217 . . . . 5  |-  U. (/)  =  (/)
3028, 29syl6eq 2521 . . . 4  |-  ( -.  C  e.  Cat  ->  U.
ran  H  =  (/) )
3117, 30eqtr4d 2508 . . 3  |-  ( -.  C  e.  Cat  ->  (Nat
`  C )  = 
U. ran  H )
3212, 31pm2.61i 169 . 2  |-  (Nat `  C )  =  U. ran  H
331, 32eqtri 2493 1  |-  A  = 
U. ran  H
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    = wceq 1452    e. wcel 1904   _Vcvv 3031   (/)c0 3722   {csn 3959   U.cuni 4190    |-> cmpt 4454    X. cxp 4837   dom cdm 4839   ran crn 4840   ` cfv 5589   Basecbs 15199   Hom chom 15279   Catccat 15648  Natcarw 15995  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  ax-un 6602
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-sbc 3256  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-id 4754  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-iota 5553  df-fun 5591  df-fv 5597  df-homa 15999  df-arw 16000
This theorem is referenced by:  arwhoma  16018  homarw  16019
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