Theorem homarw 18008

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 7395	. 2 ⊢ (𝑋𝐻𝑌) ⊆ ∪ ran 𝐻
2		arwrcl.a	. . 3 ⊢ 𝐴 = (Arrow‘𝐶)
3		arwhoma.h	. . 3 ⊢ 𝐻 = (Homa‘𝐶)
4	2, 3	arwval 18005	. 2 ⊢ 𝐴 = ∪ ran 𝐻
5	1, 4	sseqtrri 3965	1 ⊢ (𝑋𝐻𝑌) ⊆ 𝐴

Syntax hints:   = wceq 1548   ⊆ wss 3884  ∪ cuni 4840  ran crn 5621  ‘cfv 6488  (class class class)co 7359  Arrowcarw 17984  Hom_achoma 17985

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1975  ax-7 2016  ax-8 2123  ax-9 2131  ax-10 2154  ax-11 2170  ax-12 2191  ax-ext 2713  ax-sep 5220  ax-nul 5230  ax-pr 5364  ax-un 7681

This theorem depends on definitions:  df-bi 209  df-an 398  df-or 855  df-3an 1095  df-tru 1551  df-fal 1561  df-ex 1788  df-nf 1792  df-sb 2075  df-mo 2545  df-eu 2575  df-clab 2720  df-cleq 2733  df-clel 2816  df-nfc 2890  df-ne 2937  df-ral 3056  df-rex 3066  df-rab 3394  df-v 3435  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4264  df-if 4457  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4841  df-br 5075  df-opab 5137  df-mpt 5156  df-id 5515  df-xp 5626  df-rel 5627  df-cnv 5628  df-co 5629  df-dm 5630  df-rn 5631  df-iota 6444  df-fun 6490  df-fv 6496  df-ov 7362  df-homa 17988  df-arw 17989

This theorem is referenced by:  idaf  18025  homdmcoa  18029  coaval  18030  coapm  18033  termcarweu  50030  arweuthinc  50031  arweutermc  50032