homacd - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > homacd		Structured version Visualization version GIF version

Theorem homacd 18086

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 18070	. . . 4 ⊢ cod_a = (2^nd ∘ 1^st )
2	1	fveq1i 6907	. . 3 ⊢ (cod_a‘𝐹) = ((2^nd ∘ 1^st )‘𝐹)
3		fo1st 8034	. . . . 5 ⊢ 1^st :V–onto→V
4		fof 6820	. . . . 5 ⊢ (1^st :V–onto→V → 1^st :V⟶V)
5	3, 4	ax-mp 5	. . . 4 ⊢ 1^st :V⟶V
6		elex 3501	. . . 4 ⊢ (𝐹 ∈ (𝑋𝐻𝑌) → 𝐹 ∈ V)
7		fvco3 7008	. . . 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 2789	. 2 ⊢ (𝐹 ∈ (𝑋𝐻𝑌) → (cod_a‘𝐹) = (2^nd ‘(1^st ‘𝐹)))
10		homahom.h	. . . . . 6 ⊢ 𝐻 = (Hom_a‘𝐶)
11	10	homarel 18081	. . . . 5 ⊢ Rel (𝑋𝐻𝑌)
12		1st2ndbr 8067	. . . . 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 18082	. . . 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 6910	. 2 ⊢ (𝐹 ∈ (𝑋𝐻𝑌) → (2^nd ‘(1^st ‘𝐹)) = (2^nd ‘⟨𝑋, 𝑌⟩))
17		eqid 2737	. . . 4 ⊢ (Base‘𝐶) = (Base‘𝐶)
18	10, 17	homarcl2 18080	. . 3 ⊢ (𝐹 ∈ (𝑋𝐻𝑌) → (𝑋 ∈ (Base‘𝐶) ∧ 𝑌 ∈ (Base‘𝐶)))
19		op2ndg 8027	. . 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 2781	1 ⊢ (𝐹 ∈ (𝑋𝐻𝑌) → (cod_a‘𝐹) = 𝑌)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2108 Vcvv 3480 ⟨cop 4632 class class class wbr 5143 ∘ ccom 5689 Rel wrel 5690 ⟶wf 6557 –onto→wfo 6559 ‘cfv 6561 (class class class)co 7431 1^st c1st 8012 2^nd c2nd 8013 Basecbs 17247 cod_accoda 18066 Hom_achoma 18068

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-ov 7434 df-1st 8014 df-2nd 8015 df-coda 18070 df-homa 18071

This theorem is referenced by: arwhoma 18090 idacd 18107 homdmcoa 18112 coaval 18113

W3C validator