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Theorem arwval 15449
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 5872 . . . . . . 7  |-  ( c  =  C  ->  (Homa `  c
)  =  (Homa `  C
) )
3 arwval.h . . . . . . 7  |-  H  =  (Homa
`  C )
42, 3syl6eqr 2516 . . . . . 6  |-  ( c  =  C  ->  (Homa `  c
)  =  H )
54rneqd 5240 . . . . 5  |-  ( c  =  C  ->  ran  (Homa `  c )  =  ran  H )
65unieqd 4261 . . . 4  |-  ( c  =  C  ->  U. ran  (Homa `  c )  =  U. ran  H )
7 df-arw 15433 . . . 4  |- Nat  =  ( c  e.  Cat  |->  U.
ran  (Homa
`  c ) )
8 fvex 5882 . . . . . . 7  |-  (Homa `  C
)  e.  _V
93, 8eqeltri 2541 . . . . . 6  |-  H  e. 
_V
109rnex 6733 . . . . 5  |-  ran  H  e.  _V
1110uniex 6595 . . . 4  |-  U. ran  H  e.  _V
126, 7, 11fvmpt 5956 . . 3  |-  ( C  e.  Cat  ->  (Nat `  C )  =  U. ran  H )
137dmmptss 5509 . . . . . . 7  |-  dom Nat  C_  Cat
1413sseli 3495 . . . . . 6  |-  ( C  e.  dom Nat  ->  C  e. 
Cat )
1514con3i 135 . . . . 5  |-  ( -.  C  e.  Cat  ->  -.  C  e.  dom Nat )
16 ndmfv 5896 . . . . 5  |-  ( -.  C  e.  dom Nat  ->  (Nat
`  C )  =  (/) )
1715, 16syl 16 . . . 4  |-  ( -.  C  e.  Cat  ->  (Nat
`  C )  =  (/) )
18 df-homa 15432 . . . . . . . . . . . . 13  |- Homa  =  ( c  e.  Cat  |->  ( x  e.  ( ( Base `  c
)  X.  ( Base `  c ) )  |->  ( { x }  X.  ( ( Hom  `  c
) `  x )
) ) )
1918dmmptss 5509 . . . . . . . . . . . 12  |-  dom Homa  C_  Cat
2019sseli 3495 . . . . . . . . . . 11  |-  ( C  e.  dom Homa  ->  C  e.  Cat )
2120con3i 135 . . . . . . . . . 10  |-  ( -.  C  e.  Cat  ->  -.  C  e.  dom Homa )
22 ndmfv 5896 . . . . . . . . . 10  |-  ( -.  C  e.  dom Homa  ->  (Homa
`  C )  =  (/) )
2321, 22syl 16 . . . . . . . . 9  |-  ( -.  C  e.  Cat  ->  (Homa `  C )  =  (/) )
243, 23syl5eq 2510 . . . . . . . 8  |-  ( -.  C  e.  Cat  ->  H  =  (/) )
2524rneqd 5240 . . . . . . 7  |-  ( -.  C  e.  Cat  ->  ran 
H  =  ran  (/) )
26 rn0 5264 . . . . . . 7  |-  ran  (/)  =  (/)
2725, 26syl6eq 2514 . . . . . 6  |-  ( -.  C  e.  Cat  ->  ran 
H  =  (/) )
2827unieqd 4261 . . . . 5  |-  ( -.  C  e.  Cat  ->  U.
ran  H  =  U. (/) )
29 uni0 4278 . . . . 5  |-  U. (/)  =  (/)
3028, 29syl6eq 2514 . . . 4  |-  ( -.  C  e.  Cat  ->  U.
ran  H  =  (/) )
3117, 30eqtr4d 2501 . . 3  |-  ( -.  C  e.  Cat  ->  (Nat
`  C )  = 
U. ran  H )
3212, 31pm2.61i 164 . 2  |-  (Nat `  C )  =  U. ran  H
331, 32eqtri 2486 1  |-  A  = 
U. ran  H
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    = wceq 1395    e. wcel 1819   _Vcvv 3109   (/)c0 3793   {csn 4032   U.cuni 4251    |-> cmpt 4515    X. cxp 5006   dom cdm 5008   ran crn 5009   ` cfv 5594   Basecbs 14644   Hom chom 14723   Catccat 15081  Natcarw 15428  Homachoma 15429
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-sbc 3328  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-br 4457  df-opab 4516  df-mpt 4517  df-id 4804  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fv 5602  df-homa 15432  df-arw 15433
This theorem is referenced by:  arwhoma  15451  homarw  15452
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