Theorem homahom 18175

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 18172	. . 3 ⊢ Rel (𝑋𝐻𝑌)
3		1st2ndbr 8036	. . 3 ⊢ ((Rel (𝑋𝐻𝑌) ∧ 𝐹 ∈ (𝑋𝐻𝑌)) → (1st ‘𝐹)(𝑋𝐻𝑌)(2nd ‘𝐹))
4	2, 3	mpan 703	. 2 ⊢ (𝐹 ∈ (𝑋𝐻𝑌) → (1st ‘𝐹)(𝑋𝐻𝑌)(2nd ‘𝐹))
5		homahom.j	. . 3 ⊢ 𝐽 = (Hom ‘𝐶)
6	1, 5	homahom2 18174	. 2 ⊢ ((1st ‘𝐹)(𝑋𝐻𝑌)(2nd ‘𝐹) → (2nd ‘𝐹) ∈ (𝑋𝐽𝑌))
7	4, 6	syl 18	1 ⊢ (𝐹 ∈ (𝑋𝐻𝑌) → (2nd ‘𝐹) ∈ (𝑋𝐽𝑌))

This proof depends on syntax axioms:   → wi 4   = wceq 1570   ∈ wcel 2145   class class class wbr 5102  Rel wrel 5652  ‘cfv 6527  (class class class)co 7408  1^st c1st 7982  2^nd c2nd 7983  Hom chom 17400  Hom_achoma 18159

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7734

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-id 5542  df-xp 5653  df-rel 5654  df-cnv 5655  df-co 5656  df-dm 5657  df-rn 5658  df-res 5659  df-ima 5660  df-iota 6483  df-fun 6529  df-fn 6530  df-f 6531  df-f1 6532  df-fo 6533  df-f1o 6534  df-fv 6535  df-ov 7411  df-1st 7984  df-2nd 7985  df-homa 18162

This theorem is used by:  arwhom  18187  coahom  18206  arwlid  18208  arwrid  18209  arwass  18210