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 7423	. 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 3996	1 ⊢ (𝑋𝐻𝑌) ⊆ 𝐴

Syntax hints:   = wceq 1540   ⊆ wss 3914  ∪ cuni 4871  ran crn 5639  ‘cfv 6511  (class class class)co 7387  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 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 2701  ax-sep 5251  ax-nul 5261  ax-pr 5387  ax-un 7711

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 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3406  df-v 3449  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4297  df-if 4489  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-br 5108  df-opab 5170  df-mpt 5189  df-id 5533  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-iota 6464  df-fun 6513  df-fv 6519  df-ov 7390  df-homa 17988  df-arw 17989

This theorem is referenced by:  idaf  18025  homdmcoa  18029  coaval  18030  coapm  18033  termcarweu  49517  arweuthinc  49518  arweutermc  49519