Theorem homarw 18013

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 7403	. 2 ⊢ (𝑋𝐻𝑌) ⊆ ∪ ran 𝐻
2		arwrcl.a	. . 3 ⊢ 𝐴 = (Arrow‘𝐶)
3		arwhoma.h	. . 3 ⊢ 𝐻 = (Homa‘𝐶)
4	2, 3	arwval 18010	. 2 ⊢ 𝐴 = ∪ ran 𝐻
5	1, 4	sseqtrri 3971	1 ⊢ (𝑋𝐻𝑌) ⊆ 𝐴

Syntax hints:   = wceq 1542   ⊆ wss 3889  ∪ cuni 4850  ran crn 5632  ‘cfv 6498  (class class class)co 7367  Arrowcarw 17989  Hom_achoma 17990

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708  ax-sep 5231  ax-nul 5241  ax-pr 5375  ax-un 7689

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3062  df-rab 3390  df-v 3431  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4274  df-if 4467  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-br 5086  df-opab 5148  df-mpt 5167  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-iota 6454  df-fun 6500  df-fv 6506  df-ov 7370  df-homa 17993  df-arw 17994

This theorem is referenced by:  idaf  18030  homdmcoa  18034  coaval  18035  coapm  18038  termcarweu  50003  arweuthinc  50004  arweutermc  50005