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Theorem homarw 14203
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 6109 . 2  |-  ( X H Y )  C_  U.
ran  H
2 arwrcl.a . . 3  |-  A  =  (Nat `  C )
3 arwhoma.h . . 3  |-  H  =  (Homa
`  C )
42, 3arwval 14200 . 2  |-  A  = 
U. ran  H
51, 4sseqtr4i 3383 1  |-  ( X H Y )  C_  A
Colors of variables: wff set class
Syntax hints:    = wceq 1653    C_ wss 3322   U.cuni 4017   ran crn 4881   ` cfv 5456  (class class class)co 6083  Natcarw 14179  Homachoma 14180
This theorem is referenced by:  idaf  14220  homdmcoa  14224  coaval  14225  coapm  14228
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-ov 6086  df-homa 14183  df-arw 14184
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