Theorem homarw 18064

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 = (Homa‘𝐶)

Assertion

Ref	Expression
homarw	⊢ (𝑋𝐻𝑌) ⊆ 𝐴

Proof of Theorem homarw

Step	Hyp	Ref	Expression
1		ovssunirn 7446	. 2 ⊢ (𝑋𝐻𝑌) ⊆ ∪ ran 𝐻
2		arwrcl.a	. . 3 ⊢ 𝐴 = (Arrow‘𝐶)
3		arwhoma.h	. . 3 ⊢ 𝐻 = (Homa‘𝐶)
4	2, 3	arwval 18061	. 2 ⊢ 𝐴 = ∪ ran 𝐻
5	1, 4	sseqtrri 4013	1 ⊢ (𝑋𝐻𝑌) ⊆ 𝐴

Syntax hints:   = wceq 1540   ⊆ wss 3931  ∪ cuni 4888  ran crn 5660  ‘cfv 6536  (class class class)co 7410  Arrowcarw 18040  Hom_achoma 18041

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2708  ax-sep 5271  ax-nul 5281  ax-pr 5407  ax-un 7734

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2810  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3062  df-rab 3421  df-v 3466  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-nul 4314  df-if 4506  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4889  df-br 5125  df-opab 5187  df-mpt 5207  df-id 5553  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-iota 6489  df-fun 6538  df-fv 6544  df-ov 7413  df-homa 18044  df-arw 18045

This theorem is referenced by:  idaf  18081  homdmcoa  18085  coaval  18086  coapm  18089  termcarweu  49380  arweuthinc  49381  arweutermc  49382