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Theorem arwval 14200
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 5730 . . . . . . 7  |-  ( c  =  C  ->  (Homa `  c
)  =  (Homa `  C
) )
3 arwval.h . . . . . . 7  |-  H  =  (Homa
`  C )
42, 3syl6eqr 2488 . . . . . 6  |-  ( c  =  C  ->  (Homa `  c
)  =  H )
54rneqd 5099 . . . . 5  |-  ( c  =  C  ->  ran  (Homa `  c )  =  ran  H )
65unieqd 4028 . . . 4  |-  ( c  =  C  ->  U. ran  (Homa `  c )  =  U. ran  H )
7 df-arw 14184 . . . 4  |- Nat  =  ( c  e.  Cat  |->  U.
ran  (Homa
`  c ) )
8 fvex 5744 . . . . . . 7  |-  (Homa `  C
)  e.  _V
93, 8eqeltri 2508 . . . . . 6  |-  H  e. 
_V
109rnex 5135 . . . . 5  |-  ran  H  e.  _V
1110uniex 4707 . . . 4  |-  U. ran  H  e.  _V
126, 7, 11fvmpt 5808 . . 3  |-  ( C  e.  Cat  ->  (Nat `  C )  =  U. ran  H )
137dmmptss 5368 . . . . . . 7  |-  dom Nat  C_  Cat
1413sseli 3346 . . . . . 6  |-  ( C  e.  dom Nat  ->  C  e. 
Cat )
1514con3i 130 . . . . 5  |-  ( -.  C  e.  Cat  ->  -.  C  e.  dom Nat )
16 ndmfv 5757 . . . . 5  |-  ( -.  C  e.  dom Nat  ->  (Nat
`  C )  =  (/) )
1715, 16syl 16 . . . 4  |-  ( -.  C  e.  Cat  ->  (Nat
`  C )  =  (/) )
18 df-homa 14183 . . . . . . . . . . . . 13  |- Homa  =  ( c  e.  Cat  |->  ( x  e.  ( ( Base `  c
)  X.  ( Base `  c ) )  |->  ( { x }  X.  ( (  Hom  `  c
) `  x )
) ) )
1918dmmptss 5368 . . . . . . . . . . . 12  |-  dom Homa  C_  Cat
2019sseli 3346 . . . . . . . . . . 11  |-  ( C  e.  dom Homa  ->  C  e.  Cat )
2120con3i 130 . . . . . . . . . 10  |-  ( -.  C  e.  Cat  ->  -.  C  e.  dom Homa )
22 ndmfv 5757 . . . . . . . . . 10  |-  ( -.  C  e.  dom Homa  ->  (Homa
`  C )  =  (/) )
2321, 22syl 16 . . . . . . . . 9  |-  ( -.  C  e.  Cat  ->  (Homa `  C )  =  (/) )
243, 23syl5eq 2482 . . . . . . . 8  |-  ( -.  C  e.  Cat  ->  H  =  (/) )
2524rneqd 5099 . . . . . . 7  |-  ( -.  C  e.  Cat  ->  ran 
H  =  ran  (/) )
26 rn0 5129 . . . . . . 7  |-  ran  (/)  =  (/)
2725, 26syl6eq 2486 . . . . . 6  |-  ( -.  C  e.  Cat  ->  ran 
H  =  (/) )
2827unieqd 4028 . . . . 5  |-  ( -.  C  e.  Cat  ->  U.
ran  H  =  U. (/) )
29 uni0 4044 . . . . 5  |-  U. (/)  =  (/)
3028, 29syl6eq 2486 . . . 4  |-  ( -.  C  e.  Cat  ->  U.
ran  H  =  (/) )
3117, 30eqtr4d 2473 . . 3  |-  ( -.  C  e.  Cat  ->  (Nat
`  C )  = 
U. ran  H )
3212, 31pm2.61i 159 . 2  |-  (Nat `  C )  =  U. ran  H
331, 32eqtri 2458 1  |-  A  = 
U. ran  H
Colors of variables: wff set class
Syntax hints:   -. wn 3    = wceq 1653    e. wcel 1726   _Vcvv 2958   (/)c0 3630   {csn 3816   U.cuni 4017    e. cmpt 4268    X. cxp 4878   dom cdm 4880   ran crn 4881   ` cfv 5456   Basecbs 13471    Hom chom 13542   Catccat 13891  Natcarw 14179  Homachoma 14180
This theorem is referenced by:  arwhoma  14202  homarw  14203
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  ax-un 4703
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-sbc 3164  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-id 4500  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fv 5464  df-homa 14183  df-arw 14184
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