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Theorem homarw 16465
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 = (Arrow‘𝐶)
arwhoma.h 𝐻 = (Homa𝐶)
Assertion
Ref Expression
homarw (𝑋𝐻𝑌) ⊆ 𝐴

Proof of Theorem homarw
StepHypRef Expression
1 ovssunirn 6557 . 2 (𝑋𝐻𝑌) ⊆ ran 𝐻
2 arwrcl.a . . 3 𝐴 = (Arrow‘𝐶)
3 arwhoma.h . . 3 𝐻 = (Homa𝐶)
42, 3arwval 16462 . 2 𝐴 = ran 𝐻
51, 4sseqtr4i 3600 1 (𝑋𝐻𝑌) ⊆ 𝐴
Colors of variables: wff setvar class
Syntax hints:   = wceq 1474  wss 3539   cuni 4366  ran crn 5029  cfv 5790  (class class class)co 6527  Arrowcarw 16441  Homachoma 16442
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2032  ax-13 2232  ax-ext 2589  ax-sep 4703  ax-nul 4712  ax-pow 4764  ax-pr 4828  ax-un 6824
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ne 2781  df-ral 2900  df-rex 2901  df-rab 2904  df-v 3174  df-sbc 3402  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-nul 3874  df-if 4036  df-sn 4125  df-pr 4127  df-op 4131  df-uni 4367  df-br 4578  df-opab 4638  df-mpt 4639  df-id 4943  df-xp 5034  df-rel 5035  df-cnv 5036  df-co 5037  df-dm 5038  df-rn 5039  df-res 5040  df-ima 5041  df-iota 5754  df-fun 5792  df-fv 5798  df-ov 6530  df-homa 16445  df-arw 16446
This theorem is referenced by:  idaf  16482  homdmcoa  16486  coaval  16487  coapm  16490
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