Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  homahom Structured version   Visualization version   GIF version

Theorem homahom 16736
 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 16733 . . 3 Rel (𝑋𝐻𝑌)
3 1st2ndbr 7261 . . 3 ((Rel (𝑋𝐻𝑌) ∧ 𝐹 ∈ (𝑋𝐻𝑌)) → (1st𝐹)(𝑋𝐻𝑌)(2nd𝐹))
42, 3mpan 706 . 2 (𝐹 ∈ (𝑋𝐻𝑌) → (1st𝐹)(𝑋𝐻𝑌)(2nd𝐹))
5 homahom.j . . 3 𝐽 = (Hom ‘𝐶)
61, 5homahom2 16735 . 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 1523   ∈ wcel 2030   class class class wbr 4685  Rel wrel 5148  ‘cfv 5926  (class class class)co 6690  1st c1st 7208  2nd c2nd 7209  Hom chom 15999  Homachoma 16720 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-ov 6693  df-1st 7210  df-2nd 7211  df-homa 16723 This theorem is referenced by:  arwhom  16748  coahom  16767  arwlid  16769  arwrid  16770  arwass  16771
 Copyright terms: Public domain W3C validator