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 = (Homa‘𝐶)
homahom.j	⊢ 𝐽 = (Hom ‘𝐶)

Assertion

Ref	Expression
homahom	⊢ (𝐹 ∈ (𝑋𝐻𝑌) → (2nd ‘𝐹) ∈ (𝑋𝐽𝑌))

Proof of Theorem homahom

Step	Hyp	Ref	Expression
1		homahom.h	. . . 4 ⊢ 𝐻 = (Homa‘𝐶)
2	1	homarel 17376	. . 3 ⊢ Rel (𝑋𝐻𝑌)
3		1st2ndbr 7751	. . 3 ⊢ ((Rel (𝑋𝐻𝑌) ∧ 𝐹 ∈ (𝑋𝐻𝑌)) → (1st ‘𝐹)(𝑋𝐻𝑌)(2nd ‘𝐹))
4	2, 3	mpan 689	. 2 ⊢ (𝐹 ∈ (𝑋𝐻𝑌) → (1st ‘𝐹)(𝑋𝐻𝑌)(2nd ‘𝐹))
5		homahom.j	. . 3 ⊢ 𝐽 = (Hom ‘𝐶)
6	1, 5	homahom2 17378	. 2 ⊢ ((1st ‘𝐹)(𝑋𝐻𝑌)(2nd ‘𝐹) → (2nd ‘𝐹) ∈ (𝑋𝐽𝑌))
7	4, 6	syl 17	1 ⊢ (𝐹 ∈ (𝑋𝐻𝑌) → (2nd ‘𝐹) ∈ (𝑋𝐽𝑌))

Syntax hints:   → wi 4   = wceq 1538   ∈ wcel 2111   class class class wbr 5036  Rel wrel 5533  ‘cfv 6340  (class class class)co 7156  1^st c1st 7697  2^nd c2nd 7698  Hom chom 16648  Hom_achoma 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