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Theorem homarw 15028
Description: A hom-set is a subset of the collection of all arrows. (Contributed by Mario Carneiro, 11-Jan-2017.)
Hypotheses
Ref Expression
arwrcl.a  |-  A  =  (Nat `  C )
arwhoma.h  |-  H  =  (Homa
`  C )
Assertion
Ref Expression
homarw  |-  ( X H Y )  C_  A

Proof of Theorem homarw
StepHypRef Expression
1 ovssunirn 6221 . 2  |-  ( X H Y )  C_  U.
ran  H
2 arwrcl.a . . 3  |-  A  =  (Nat `  C )
3 arwhoma.h . . 3  |-  H  =  (Homa
`  C )
42, 3arwval 15025 . 2  |-  A  = 
U. ran  H
51, 4sseqtr4i 3492 1  |-  ( X H Y )  C_  A
Colors of variables: wff setvar class
Syntax hints:    = wceq 1370    C_ wss 3431   U.cuni 4194   ran crn 4944   ` cfv 5521  (class class class)co 6195  Natcarw 15004  Homachoma 15005
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1954  ax-ext 2431  ax-sep 4516  ax-nul 4524  ax-pow 4573  ax-pr 4634  ax-un 6477
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2265  df-mo 2266  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2602  df-ne 2647  df-ral 2801  df-rex 2802  df-rab 2805  df-v 3074  df-sbc 3289  df-dif 3434  df-un 3436  df-in 3438  df-ss 3445  df-nul 3741  df-if 3895  df-sn 3981  df-pr 3983  df-op 3987  df-uni 4195  df-br 4396  df-opab 4454  df-mpt 4455  df-id 4739  df-xp 4949  df-rel 4950  df-cnv 4951  df-co 4952  df-dm 4953  df-rn 4954  df-res 4955  df-ima 4956  df-iota 5484  df-fun 5523  df-fv 5529  df-ov 6198  df-homa 15008  df-arw 15009
This theorem is referenced by:  idaf  15045  homdmcoa  15049  coaval  15050  coapm  15053
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