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Theorem homarw 18035
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 7456 . 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 18032 . 2 𝐴 = βˆͺ ran 𝐻
51, 4sseqtrri 4017 1 (π‘‹π»π‘Œ) βŠ† 𝐴
Colors of variables: wff setvar class
Syntax hints:   = wceq 1534   βŠ† wss 3947  βˆͺ cuni 4908  ran crn 5679  β€˜cfv 6548  (class class class)co 7420  Arrowcarw 18011  Homachoma 18012
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-sep 5299  ax-nul 5306  ax-pr 5429  ax-un 7740
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3430  df-v 3473  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4909  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-iota 6500  df-fun 6550  df-fv 6556  df-ov 7423  df-homa 18015  df-arw 18016
This theorem is referenced by:  idaf  18052  homdmcoa  18056  coaval  18057  coapm  18060
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