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Theorem homarw 15442
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 6225 . 2  |-  ( X H Y )  C_  U.
ran  H
2 arwrcl.a . . 3  |-  A  =  (Nat `  C )
3 arwhoma.h . . 3  |-  H  =  (Homa
`  C )
42, 3arwval 15439 . 2  |-  A  = 
U. ran  H
51, 4sseqtr4i 3450 1  |-  ( X H Y )  C_  A
Colors of variables: wff setvar class
Syntax hints:    = wceq 1399    C_ wss 3389   U.cuni 4163   ran crn 4914   ` cfv 5496  (class class class)co 6196  Natcarw 15418  Homachoma 15419
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-8 1828  ax-9 1830  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006  ax-ext 2360  ax-sep 4488  ax-nul 4496  ax-pow 4543  ax-pr 4601  ax-un 6491
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1402  df-ex 1621  df-nf 1625  df-sb 1748  df-eu 2222  df-mo 2223  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2532  df-ne 2579  df-ral 2737  df-rex 2738  df-rab 2741  df-v 3036  df-sbc 3253  df-dif 3392  df-un 3394  df-in 3396  df-ss 3403  df-nul 3712  df-if 3858  df-sn 3945  df-pr 3947  df-op 3951  df-uni 4164  df-br 4368  df-opab 4426  df-mpt 4427  df-id 4709  df-xp 4919  df-rel 4920  df-cnv 4921  df-co 4922  df-dm 4923  df-rn 4924  df-res 4925  df-ima 4926  df-iota 5460  df-fun 5498  df-fv 5504  df-ov 6199  df-homa 15422  df-arw 15423
This theorem is referenced by:  idaf  15459  homdmcoa  15463  coaval  15464  coapm  15467
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