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Theorem homacd 17956

Description: The codomain of an arrow with known domain and codomain. (Contributed by Mario Carneiro, 11-Jan-2017.)

**Hypothesis**
Ref	Expression
homahom.h	⊢ 𝐻 = (Hom_a‘𝐶)

**Assertion**
Ref	Expression
homacd	⊢ (𝐹 ∈ (𝑋𝐻𝑌) → (cod_a‘𝐹) = 𝑌)

**Proof of Theorem homacd**
Step	Hyp	Ref	Expression
1		df-coda 17940	. . . 4 ⊢ cod_a = (2^nd ∘ 1^st )
2	1	fveq1i 6832	. . 3 ⊢ (cod_a‘𝐹) = ((2^nd ∘ 1^st )‘𝐹)
3		fo1st 7950	. . . . 5 ⊢ 1^st :V–onto→V
4		fof 6743	. . . . 5 ⊢ (1^st :V–onto→V → 1^st :V⟶V)
5	3, 4	ax-mp 5	. . . 4 ⊢ 1^st :V⟶V
6		elex 3458	. . . 4 ⊢ (𝐹 ∈ (𝑋𝐻𝑌) → 𝐹 ∈ V)
7		fvco3 6930	. . . 4 ⊢ ((1^st :V⟶V ∧ 𝐹 ∈ V) → ((2^nd ∘ 1^st )‘𝐹) = (2^nd ‘(1^st ‘𝐹)))
8	5, 6, 7	sylancr 587	. . 3 ⊢ (𝐹 ∈ (𝑋𝐻𝑌) → ((2^nd ∘ 1^st )‘𝐹) = (2^nd ‘(1^st ‘𝐹)))
9	2, 8	eqtrid 2780	. 2 ⊢ (𝐹 ∈ (𝑋𝐻𝑌) → (cod_a‘𝐹) = (2^nd ‘(1^st ‘𝐹)))
10		homahom.h	. . . . . 6 ⊢ 𝐻 = (Hom_a‘𝐶)
11	10	homarel 17951	. . . . 5 ⊢ Rel (𝑋𝐻𝑌)
12		1st2ndbr 7983	. . . . 5 ⊢ ((Rel (𝑋𝐻𝑌) ∧ 𝐹 ∈ (𝑋𝐻𝑌)) → (1^st ‘𝐹)(𝑋𝐻𝑌)(2^nd ‘𝐹))
13	11, 12	mpan 690	. . . 4 ⊢ (𝐹 ∈ (𝑋𝐻𝑌) → (1^st ‘𝐹)(𝑋𝐻𝑌)(2^nd ‘𝐹))
14	10	homa1 17952	. . . 4 ⊢ ((1^st ‘𝐹)(𝑋𝐻𝑌)(2^nd ‘𝐹) → (1^st ‘𝐹) = ⟨𝑋, 𝑌⟩)
15	13, 14	syl 17	. . 3 ⊢ (𝐹 ∈ (𝑋𝐻𝑌) → (1^st ‘𝐹) = ⟨𝑋, 𝑌⟩)
16	15	fveq2d 6835	. 2 ⊢ (𝐹 ∈ (𝑋𝐻𝑌) → (2^nd ‘(1^st ‘𝐹)) = (2^nd ‘⟨𝑋, 𝑌⟩))
17		eqid 2733	. . . 4 ⊢ (Base‘𝐶) = (Base‘𝐶)
18	10, 17	homarcl2 17950	. . 3 ⊢ (𝐹 ∈ (𝑋𝐻𝑌) → (𝑋 ∈ (Base‘𝐶) ∧ 𝑌 ∈ (Base‘𝐶)))
19		op2ndg 7943	. . 3 ⊢ ((𝑋 ∈ (Base‘𝐶) ∧ 𝑌 ∈ (Base‘𝐶)) → (2^nd ‘⟨𝑋, 𝑌⟩) = 𝑌)
20	18, 19	syl 17	. 2 ⊢ (𝐹 ∈ (𝑋𝐻𝑌) → (2^nd ‘⟨𝑋, 𝑌⟩) = 𝑌)
21	9, 16, 20	3eqtrd 2772	1 ⊢ (𝐹 ∈ (𝑋𝐻𝑌) → (cod_a‘𝐹) = 𝑌)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1541 ∈ wcel 2113 Vcvv 3437 ⟨cop 4583 class class class wbr 5095 ∘ ccom 5625 Rel wrel 5626 ⟶wf 6485 –onto→wfo 6487 ‘cfv 6489 (class class class)co 7355 1^st c1st 7928 2^nd c2nd 7929 Basecbs 17127 cod_accoda 17936 Hom_achoma 17938

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2115 ax-9 2123 ax-10 2146 ax-11 2162 ax-12 2182 ax-ext 2705 ax-rep 5221 ax-sep 5238 ax-nul 5248 ax-pow 5307 ax-pr 5374 ax-un 7677

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2725 df-clel 2808 df-nfc 2882 df-ne 2930 df-ral 3049 df-rex 3058 df-reu 3348 df-rab 3397 df-v 3439 df-sbc 3738 df-csb 3847 df-dif 3901 df-un 3903 df-in 3905 df-ss 3915 df-nul 4283 df-if 4477 df-pw 4553 df-sn 4578 df-pr 4580 df-op 4584 df-uni 4861 df-iun 4945 df-br 5096 df-opab 5158 df-mpt 5177 df-id 5516 df-xp 5627 df-rel 5628 df-cnv 5629 df-co 5630 df-dm 5631 df-rn 5632 df-res 5633 df-ima 5634 df-iota 6445 df-fun 6491 df-fn 6492 df-f 6493 df-f1 6494 df-fo 6495 df-f1o 6496 df-fv 6497 df-ov 7358 df-1st 7930 df-2nd 7931 df-coda 17940 df-homa 17941

This theorem is referenced by: arwhoma 17960 idacd 17977 homdmcoa 17982 coaval 17983

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