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Theorem homarw 18004
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 = (Arrowβ€˜πΆ)
arwhoma.h 𝐻 = (Homaβ€˜πΆ)
Assertion
Ref Expression
homarw (π‘‹π»π‘Œ) βŠ† 𝐴

Proof of Theorem homarw
StepHypRef Expression
1 ovssunirn 7438 . 2 (π‘‹π»π‘Œ) βŠ† βˆͺ ran 𝐻
2 arwrcl.a . . 3 𝐴 = (Arrowβ€˜πΆ)
3 arwhoma.h . . 3 𝐻 = (Homaβ€˜πΆ)
42, 3arwval 18001 . 2 𝐴 = βˆͺ ran 𝐻
51, 4sseqtrri 4012 1 (π‘‹π»π‘Œ) βŠ† 𝐴
Colors of variables: wff setvar class
Syntax hints:   = wceq 1533   βŠ† wss 3941  βˆͺ cuni 4900  ran crn 5668  β€˜cfv 6534  (class class class)co 7402  Arrowcarw 17980  Homachoma 17981
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-sep 5290  ax-nul 5297  ax-pr 5418  ax-un 7719
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-ral 3054  df-rex 3063  df-rab 3425  df-v 3468  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4316  df-if 4522  df-sn 4622  df-pr 4624  df-op 4628  df-uni 4901  df-br 5140  df-opab 5202  df-mpt 5223  df-id 5565  df-xp 5673  df-rel 5674  df-cnv 5675  df-co 5676  df-dm 5677  df-rn 5678  df-iota 6486  df-fun 6536  df-fv 6542  df-ov 7405  df-homa 17984  df-arw 17985
This theorem is referenced by:  idaf  18021  homdmcoa  18025  coaval  18026  coapm  18029
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