Theorem homahom 17670

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 17667	. . 3 ⊢ Rel (𝑋𝐻𝑌)
3		1st2ndbr 7856	. . 3 ⊢ ((Rel (𝑋𝐻𝑌) ∧ 𝐹 ∈ (𝑋𝐻𝑌)) → (1st ‘𝐹)(𝑋𝐻𝑌)(2nd ‘𝐹))
4	2, 3	mpan 686	. 2 ⊢ (𝐹 ∈ (𝑋𝐻𝑌) → (1st ‘𝐹)(𝑋𝐻𝑌)(2nd ‘𝐹))
5		homahom.j	. . 3 ⊢ 𝐽 = (Hom ‘𝐶)
6	1, 5	homahom2 17669	. 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 1539   ∈ wcel 2108   class class class wbr 5070  Rel wrel 5585  ‘cfv 6418  (class class class)co 7255  1^st c1st 7802  2^nd c2nd 7803  Hom chom 16899  Hom_achoma 17654

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-rep 5205  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-reu 3070  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-iun 4923  df-br 5071  df-opab 5133  df-mpt 5154  df-id 5480  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-ov 7258  df-1st 7804  df-2nd 7805  df-homa 17657

This theorem is referenced by:  arwhom  17682  coahom  17701  arwlid  17703  arwrid  17704  arwass  17705