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

Theorem homahom 17379
 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 17376 . . 3 Rel (𝑋𝐻𝑌)
3 1st2ndbr 7751 . . 3 ((Rel (𝑋𝐻𝑌) ∧ 𝐹 ∈ (𝑋𝐻𝑌)) → (1st𝐹)(𝑋𝐻𝑌)(2nd𝐹))
42, 3mpan 689 . 2 (𝐹 ∈ (𝑋𝐻𝑌) → (1st𝐹)(𝑋𝐻𝑌)(2nd𝐹))
5 homahom.j . . 3 𝐽 = (Hom ‘𝐶)
61, 5homahom2 17378 . 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 1538   ∈ wcel 2111   class class class wbr 5036  Rel wrel 5533  ‘cfv 6340  (class class class)co 7156  1st c1st 7697  2nd c2nd 7698  Hom chom 16648  Homachoma 17363 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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-rep 5160  ax-sep 5173  ax-nul 5180  ax-pow 5238  ax-pr 5302  ax-un 7465 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ne 2952  df-ral 3075  df-rex 3076  df-reu 3077  df-rab 3079  df-v 3411  df-sbc 3699  df-csb 3808  df-dif 3863  df-un 3865  df-in 3867  df-ss 3877  df-nul 4228  df-if 4424  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4802  df-iun 4888  df-br 5037  df-opab 5099  df-mpt 5117  df-id 5434  df-xp 5534  df-rel 5535  df-cnv 5536  df-co 5537  df-dm 5538  df-rn 5539  df-res 5540  df-ima 5541  df-iota 6299  df-fun 6342  df-fn 6343  df-f 6344  df-f1 6345  df-fo 6346  df-f1o 6347  df-fv 6348  df-ov 7159  df-1st 7699  df-2nd 7700  df-homa 17366 This theorem is referenced by:  arwhom  17391  coahom  17410  arwlid  17412  arwrid  17413  arwass  17414
 Copyright terms: Public domain W3C validator