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Theorem homahom 18092
Description: The second component of an arrow is the corresponding morphism (without the domain/codomain tag). (Contributed by Mario Carneiro, 11-Jan-2017.)
Hypotheses
Ref Expression
homahom.h 𝐻 = (Homa𝐶)
homahom.j 𝐽 = (Hom ‘𝐶)
Assertion
Ref Expression
homahom (𝐹 ∈ (𝑋𝐻𝑌) → (2nd𝐹) ∈ (𝑋𝐽𝑌))

Proof of Theorem homahom
StepHypRef Expression
1 homahom.h . . . 4 𝐻 = (Homa𝐶)
21homarel 18089 . . 3 Rel (𝑋𝐻𝑌)
3 1st2ndbr 8065 . . 3 ((Rel (𝑋𝐻𝑌) ∧ 𝐹 ∈ (𝑋𝐻𝑌)) → (1st𝐹)(𝑋𝐻𝑌)(2nd𝐹))
42, 3mpan 690 . 2 (𝐹 ∈ (𝑋𝐻𝑌) → (1st𝐹)(𝑋𝐻𝑌)(2nd𝐹))
5 homahom.j . . 3 𝐽 = (Hom ‘𝐶)
61, 5homahom2 18091 . 2 ((1st𝐹)(𝑋𝐻𝑌)(2nd𝐹) → (2nd𝐹) ∈ (𝑋𝐽𝑌))
74, 6syl 17 1 (𝐹 ∈ (𝑋𝐻𝑌) → (2nd𝐹) ∈ (𝑋𝐽𝑌))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1536  wcel 2105   class class class wbr 5147  Rel wrel 5693  cfv 6562  (class class class)co 7430  1st c1st 8010  2nd c2nd 8011  Hom chom 17308  Homachoma 18076
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-rep 5284  ax-sep 5301  ax-nul 5311  ax-pow 5370  ax-pr 5437  ax-un 7753
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-ral 3059  df-rex 3068  df-reu 3378  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-iun 4997  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5582  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-f1 6567  df-fo 6568  df-f1o 6569  df-fv 6570  df-ov 7433  df-1st 8012  df-2nd 8013  df-homa 18079
This theorem is referenced by:  arwhom  18104  coahom  18123  arwlid  18125  arwrid  18126  arwass  18127
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