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Theorem arwval 13875
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 5525 . . . . . . 7  |-  ( c  =  C  ->  (Homa `  c
)  =  (Homa `  C
) )
3 arwval.h . . . . . . 7  |-  H  =  (Homa
`  C )
42, 3syl6eqr 2333 . . . . . 6  |-  ( c  =  C  ->  (Homa `  c
)  =  H )
54rneqd 4906 . . . . 5  |-  ( c  =  C  ->  ran  (Homa `  c )  =  ran  H )
65unieqd 3838 . . . 4  |-  ( c  =  C  ->  U. ran  (Homa `  c )  =  U. ran  H )
7 df-arw 13859 . . . 4  |- Nat  =  ( c  e.  Cat  |->  U.
ran  (Homa
`  c ) )
8 fvex 5539 . . . . . . 7  |-  (Homa `  C
)  e.  _V
93, 8eqeltri 2353 . . . . . 6  |-  H  e. 
_V
109rnex 4942 . . . . 5  |-  ran  H  e.  _V
1110uniex 4516 . . . 4  |-  U. ran  H  e.  _V
126, 7, 11fvmpt 5602 . . 3  |-  ( C  e.  Cat  ->  (Nat `  C )  =  U. ran  H )
137dmmptss 5169 . . . . . . 7  |-  dom Nat  C_  Cat
1413sseli 3176 . . . . . 6  |-  ( C  e.  dom Nat  ->  C  e. 
Cat )
1514con3i 127 . . . . 5  |-  ( -.  C  e.  Cat  ->  -.  C  e.  dom Nat )
16 ndmfv 5552 . . . . 5  |-  ( -.  C  e.  dom Nat  ->  (Nat
`  C )  =  (/) )
1715, 16syl 15 . . . 4  |-  ( -.  C  e.  Cat  ->  (Nat
`  C )  =  (/) )
18 df-homa 13858 . . . . . . . . . . . . 13  |- Homa  =  ( c  e.  Cat  |->  ( x  e.  ( ( Base `  c
)  X.  ( Base `  c ) )  |->  ( { x }  X.  ( (  Hom  `  c
) `  x )
) ) )
1918dmmptss 5169 . . . . . . . . . . . 12  |-  dom Homa  C_  Cat
2019sseli 3176 . . . . . . . . . . 11  |-  ( C  e.  dom Homa  ->  C  e.  Cat )
2120con3i 127 . . . . . . . . . 10  |-  ( -.  C  e.  Cat  ->  -.  C  e.  dom Homa )
22 ndmfv 5552 . . . . . . . . . 10  |-  ( -.  C  e.  dom Homa  ->  (Homa
`  C )  =  (/) )
2321, 22syl 15 . . . . . . . . 9  |-  ( -.  C  e.  Cat  ->  (Homa `  C )  =  (/) )
243, 23syl5eq 2327 . . . . . . . 8  |-  ( -.  C  e.  Cat  ->  H  =  (/) )
2524rneqd 4906 . . . . . . 7  |-  ( -.  C  e.  Cat  ->  ran 
H  =  ran  (/) )
26 rn0 4936 . . . . . . 7  |-  ran  (/)  =  (/)
2725, 26syl6eq 2331 . . . . . 6  |-  ( -.  C  e.  Cat  ->  ran 
H  =  (/) )
2827unieqd 3838 . . . . 5  |-  ( -.  C  e.  Cat  ->  U.
ran  H  =  U. (/) )
29 uni0 3854 . . . . 5  |-  U. (/)  =  (/)
3028, 29syl6eq 2331 . . . 4  |-  ( -.  C  e.  Cat  ->  U.
ran  H  =  (/) )
3117, 30eqtr4d 2318 . . 3  |-  ( -.  C  e.  Cat  ->  (Nat
`  C )  = 
U. ran  H )
3212, 31pm2.61i 156 . 2  |-  (Nat `  C )  =  U. ran  H
331, 32eqtri 2303 1  |-  A  = 
U. ran  H
Colors of variables: wff set class
Syntax hints:   -. wn 3    = wceq 1623    e. wcel 1684   _Vcvv 2788   (/)c0 3455   {csn 3640   U.cuni 3827    e. cmpt 4077    X. cxp 4687   dom cdm 4689   ran crn 4690   ` cfv 5255   Basecbs 13148    Hom chom 13219   Catccat 13566  Natcarw 13854  Homachoma 13855
This theorem is referenced by:  arwhoma  13877  homarw  13878
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fv 5263  df-homa 13858  df-arw 13859
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