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Theorem arwval 15950
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 5870 . . . . . . 7  |-  ( c  =  C  ->  (Homa `  c
)  =  (Homa `  C
) )
3 arwval.h . . . . . . 7  |-  H  =  (Homa
`  C )
42, 3syl6eqr 2505 . . . . . 6  |-  ( c  =  C  ->  (Homa `  c
)  =  H )
54rneqd 5065 . . . . 5  |-  ( c  =  C  ->  ran  (Homa `  c )  =  ran  H )
65unieqd 4211 . . . 4  |-  ( c  =  C  ->  U. ran  (Homa `  c )  =  U. ran  H )
7 df-arw 15934 . . . 4  |- Nat  =  ( c  e.  Cat  |->  U.
ran  (Homa
`  c ) )
8 fvex 5880 . . . . . . 7  |-  (Homa `  C
)  e.  _V
93, 8eqeltri 2527 . . . . . 6  |-  H  e. 
_V
109rnex 6732 . . . . 5  |-  ran  H  e.  _V
1110uniex 6592 . . . 4  |-  U. ran  H  e.  _V
126, 7, 11fvmpt 5953 . . 3  |-  ( C  e.  Cat  ->  (Nat `  C )  =  U. ran  H )
137dmmptss 5334 . . . . . . 7  |-  dom Nat  C_  Cat
1413sseli 3430 . . . . . 6  |-  ( C  e.  dom Nat  ->  C  e. 
Cat )
1514con3i 141 . . . . 5  |-  ( -.  C  e.  Cat  ->  -.  C  e.  dom Nat )
16 ndmfv 5894 . . . . 5  |-  ( -.  C  e.  dom Nat  ->  (Nat
`  C )  =  (/) )
1715, 16syl 17 . . . 4  |-  ( -.  C  e.  Cat  ->  (Nat
`  C )  =  (/) )
18 df-homa 15933 . . . . . . . . . . . . 13  |- Homa  =  ( c  e.  Cat  |->  ( x  e.  ( ( Base `  c
)  X.  ( Base `  c ) )  |->  ( { x }  X.  ( ( Hom  `  c
) `  x )
) ) )
1918dmmptss 5334 . . . . . . . . . . . 12  |-  dom Homa  C_  Cat
2019sseli 3430 . . . . . . . . . . 11  |-  ( C  e.  dom Homa  ->  C  e.  Cat )
2120con3i 141 . . . . . . . . . 10  |-  ( -.  C  e.  Cat  ->  -.  C  e.  dom Homa )
22 ndmfv 5894 . . . . . . . . . 10  |-  ( -.  C  e.  dom Homa  ->  (Homa
`  C )  =  (/) )
2321, 22syl 17 . . . . . . . . 9  |-  ( -.  C  e.  Cat  ->  (Homa `  C )  =  (/) )
243, 23syl5eq 2499 . . . . . . . 8  |-  ( -.  C  e.  Cat  ->  H  =  (/) )
2524rneqd 5065 . . . . . . 7  |-  ( -.  C  e.  Cat  ->  ran 
H  =  ran  (/) )
26 rn0 5089 . . . . . . 7  |-  ran  (/)  =  (/)
2725, 26syl6eq 2503 . . . . . 6  |-  ( -.  C  e.  Cat  ->  ran 
H  =  (/) )
2827unieqd 4211 . . . . 5  |-  ( -.  C  e.  Cat  ->  U.
ran  H  =  U. (/) )
29 uni0 4228 . . . . 5  |-  U. (/)  =  (/)
3028, 29syl6eq 2503 . . . 4  |-  ( -.  C  e.  Cat  ->  U.
ran  H  =  (/) )
3117, 30eqtr4d 2490 . . 3  |-  ( -.  C  e.  Cat  ->  (Nat
`  C )  = 
U. ran  H )
3212, 31pm2.61i 168 . 2  |-  (Nat `  C )  =  U. ran  H
331, 32eqtri 2475 1  |-  A  = 
U. ran  H
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    = wceq 1446    e. wcel 1889   _Vcvv 3047   (/)c0 3733   {csn 3970   U.cuni 4201    |-> cmpt 4464    X. cxp 4835   dom cdm 4837   ran crn 4838   ` cfv 5585   Basecbs 15133   Hom chom 15213   Catccat 15582  Natcarw 15929  Homachoma 15930
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1671  ax-4 1684  ax-5 1760  ax-6 1807  ax-7 1853  ax-8 1891  ax-9 1898  ax-10 1917  ax-11 1922  ax-12 1935  ax-13 2093  ax-ext 2433  ax-sep 4528  ax-nul 4537  ax-pow 4584  ax-pr 4642  ax-un 6588
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 988  df-tru 1449  df-ex 1666  df-nf 1670  df-sb 1800  df-eu 2305  df-mo 2306  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2583  df-ne 2626  df-ral 2744  df-rex 2745  df-rab 2748  df-v 3049  df-sbc 3270  df-dif 3409  df-un 3411  df-in 3413  df-ss 3420  df-nul 3734  df-if 3884  df-sn 3971  df-pr 3973  df-op 3977  df-uni 4202  df-br 4406  df-opab 4465  df-mpt 4466  df-id 4752  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5549  df-fun 5587  df-fv 5593  df-homa 15933  df-arw 15934
This theorem is referenced by:  arwhoma  15952  homarw  15953
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