Theorem homahom 18095

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 18092	. . 3 ⊢ Rel (𝑋𝐻𝑌)
3		1st2ndbr 8038	. . 3 ⊢ ((Rel (𝑋𝐻𝑌) ∧ 𝐹 ∈ (𝑋𝐻𝑌)) → (1st ‘𝐹)(𝑋𝐻𝑌)(2nd ‘𝐹))
4	2, 3	mpan 702	. 2 ⊢ (𝐹 ∈ (𝑋𝐻𝑌) → (1st ‘𝐹)(𝑋𝐻𝑌)(2nd ‘𝐹))
5		homahom.j	. . 3 ⊢ 𝐽 = (Hom ‘𝐶)
6	1, 5	homahom2 18094	. 2 ⊢ ((1st ‘𝐹)(𝑋𝐻𝑌)(2nd ‘𝐹) → (2nd ‘𝐹) ∈ (𝑋𝐽𝑌))
7	4, 6	syl 18	1 ⊢ (𝐹 ∈ (𝑋𝐻𝑌) → (2nd ‘𝐹) ∈ (𝑋𝐽𝑌))

Syntax hints:   → wi 4   = wceq 1568   ∈ wcel 2141   class class class wbr 5108  Rel wrel 5666  ‘cfv 6536  (class class class)co 7410  1^st c1st 7983  2^nd c2nd 7984  Hom chom 17320  Hom_achoma 18079

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1823  ax-4 1837  ax-5 1938  ax-6 1995  ax-7 2036  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-rep 5237  ax-sep 5256  ax-nul 5268  ax-pow 5336  ax-pr 5404  ax-un 7732

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1571  df-fal 1581  df-ex 1808  df-nf 1812  df-sb 2095  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-reu 3368  df-rab 3415  df-v 3455  df-sbc 3744  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4487  df-pw 4563  df-sn 4589  df-pr 4591  df-op 4595  df-uni 4872  df-iun 4957  df-br 5109  df-opab 5173  df-mpt 5192  df-id 5556  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-res 5673  df-ima 5674  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-ov 7413  df-1st 7985  df-2nd 7986  df-homa 18082

This theorem is referenced by:  arwhom  18107  coahom  18126  arwlid  18128  arwrid  18129  arwass  18130