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 = (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 17130	. . 3 ⊢ Rel (𝑋𝐻𝑌)
3		1st2ndbr 7602	. . 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 17132	. 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 1522   ∈ wcel 2081   class class class wbr 4966  Rel wrel 5453  ‘cfv 6230  (class class class)co 7021  1^st c1st 7548  2^nd c2nd 7549  Hom chom 16410  Hom_achoma 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