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Theorem homahom 17133
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 17130 . . 3 Rel (𝑋𝐻𝑌)
3 1st2ndbr 7602 . . 3 ((Rel (𝑋𝐻𝑌) ∧ 𝐹 ∈ (𝑋𝐻𝑌)) → (1st𝐹)(𝑋𝐻𝑌)(2nd𝐹))
42, 3mpan 686 . 2 (𝐹 ∈ (𝑋𝐻𝑌) → (1st𝐹)(𝑋𝐻𝑌)(2nd𝐹))
5 homahom.j . . 3 𝐽 = (Hom ‘𝐶)
61, 5homahom2 17132 . 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 1522  wcel 2081   class class class wbr 4966  Rel wrel 5453  cfv 6230  (class class class)co 7021  1st c1st 7548  2nd c2nd 7549  Hom chom 16410  Homachoma 17117
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1777  ax-4 1791  ax-5 1888  ax-6 1947  ax-7 1992  ax-8 2083  ax-9 2091  ax-10 2112  ax-11 2126  ax-12 2141  ax-13 2344  ax-ext 2769  ax-rep 5086  ax-sep 5099  ax-nul 5106  ax-pow 5162  ax-pr 5226  ax-un 7324
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3an 1082  df-tru 1525  df-ex 1762  df-nf 1766  df-sb 2043  df-mo 2576  df-eu 2612  df-clab 2776  df-cleq 2788  df-clel 2863  df-nfc 2935  df-ne 2985  df-ral 3110  df-rex 3111  df-reu 3112  df-rab 3114  df-v 3439  df-sbc 3710  df-csb 3816  df-dif 3866  df-un 3868  df-in 3870  df-ss 3878  df-nul 4216  df-if 4386  df-pw 4459  df-sn 4477  df-pr 4479  df-op 4483  df-uni 4750  df-iun 4831  df-br 4967  df-opab 5029  df-mpt 5046  df-id 5353  df-xp 5454  df-rel 5455  df-cnv 5456  df-co 5457  df-dm 5458  df-rn 5459  df-res 5460  df-ima 5461  df-iota 6194  df-fun 6232  df-fn 6233  df-f 6234  df-f1 6235  df-fo 6236  df-f1o 6237  df-fv 6238  df-ov 7024  df-1st 7550  df-2nd 7551  df-homa 17120
This theorem is referenced by:  arwhom  17145  coahom  17164  arwlid  17166  arwrid  17167  arwass  17168
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