Theorem homarw 18014

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 7425	. 2 ⊢ (𝑋𝐻𝑌) ⊆ ∪ ran 𝐻
2		arwrcl.a	. . 3 ⊢ 𝐴 = (Arrow‘𝐶)
3		arwhoma.h	. . 3 ⊢ 𝐻 = (Homa‘𝐶)
4	2, 3	arwval 18011	. 2 ⊢ 𝐴 = ∪ ran 𝐻
5	1, 4	sseqtrri 3998	1 ⊢ (𝑋𝐻𝑌) ⊆ 𝐴

Syntax hints:   = wceq 1540   ⊆ wss 3916  ∪ cuni 4873  ran crn 5641  ‘cfv 6513  (class class class)co 7389  Arrowcarw 17990  Hom_achoma 17991

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 2702  ax-sep 5253  ax-nul 5263  ax-pr 5389  ax-un 7713

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 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-rab 3409  df-v 3452  df-dif 3919  df-un 3921  df-in 3923  df-ss 3933  df-nul 4299  df-if 4491  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4874  df-br 5110  df-opab 5172  df-mpt 5191  df-id 5535  df-xp 5646  df-rel 5647  df-cnv 5648  df-co 5649  df-dm 5650  df-rn 5651  df-iota 6466  df-fun 6515  df-fv 6521  df-ov 7392  df-homa 17994  df-arw 17995

This theorem is referenced by:  idaf  18031  homdmcoa  18035  coaval  18036  coapm  18039  termcarweu  49507  arweuthinc  49508  arweutermc  49509