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Theorem homacd 18001

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 17985	. . . 4 ⊢ cod_a = (2^nd ∘ 1^st )
2	1	fveq1i 6892	. . 3 ⊢ (cod_a‘𝐹) = ((2^nd ∘ 1^st )‘𝐹)
3		fo1st 7999	. . . . 5 ⊢ 1^st :V–onto→V
4		fof 6805	. . . . 5 ⊢ (1^st :V–onto→V → 1^st :V⟶V)
5	3, 4	ax-mp 5	. . . 4 ⊢ 1^st :V⟶V
6		elex 3492	. . . 4 ⊢ (𝐹 ∈ (𝑋𝐻𝑌) → 𝐹 ∈ V)
7		fvco3 6990	. . . 4 ⊢ ((1^st :V⟶V ∧ 𝐹 ∈ V) → ((2^nd ∘ 1^st )‘𝐹) = (2^nd ‘(1^st ‘𝐹)))
8	5, 6, 7	sylancr 586	. . 3 ⊢ (𝐹 ∈ (𝑋𝐻𝑌) → ((2^nd ∘ 1^st )‘𝐹) = (2^nd ‘(1^st ‘𝐹)))
9	2, 8	eqtrid 2783	. 2 ⊢ (𝐹 ∈ (𝑋𝐻𝑌) → (cod_a‘𝐹) = (2^nd ‘(1^st ‘𝐹)))
10		homahom.h	. . . . . 6 ⊢ 𝐻 = (Hom_a‘𝐶)
11	10	homarel 17996	. . . . 5 ⊢ Rel (𝑋𝐻𝑌)
12		1st2ndbr 8032	. . . . 5 ⊢ ((Rel (𝑋𝐻𝑌) ∧ 𝐹 ∈ (𝑋𝐻𝑌)) → (1^st ‘𝐹)(𝑋𝐻𝑌)(2^nd ‘𝐹))
13	11, 12	mpan 687	. . . 4 ⊢ (𝐹 ∈ (𝑋𝐻𝑌) → (1^st ‘𝐹)(𝑋𝐻𝑌)(2^nd ‘𝐹))
14	10	homa1 17997	. . . 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 6895	. 2 ⊢ (𝐹 ∈ (𝑋𝐻𝑌) → (2^nd ‘(1^st ‘𝐹)) = (2^nd ‘⟨𝑋, 𝑌⟩))
17		eqid 2731	. . . 4 ⊢ (Base‘𝐶) = (Base‘𝐶)
18	10, 17	homarcl2 17995	. . 3 ⊢ (𝐹 ∈ (𝑋𝐻𝑌) → (𝑋 ∈ (Base‘𝐶) ∧ 𝑌 ∈ (Base‘𝐶)))
19		op2ndg 7992	. . 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 2775	1 ⊢ (𝐹 ∈ (𝑋𝐻𝑌) → (cod_a‘𝐹) = 𝑌)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2105 Vcvv 3473 ⟨cop 4634 class class class wbr 5148 ∘ ccom 5680 Rel wrel 5681 ⟶wf 6539 –onto→wfo 6541 ‘cfv 6543 (class class class)co 7412 1^st c1st 7977 2^nd c2nd 7978 Basecbs 17151 cod_accoda 17981 Hom_achoma 17983

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 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7729

This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-ral 3061 df-rex 3070 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5574 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-ov 7415 df-1st 7979 df-2nd 7980 df-coda 17985 df-homa 17986

This theorem is referenced by: arwhoma 18005 idacd 18022 homdmcoa 18027 coaval 18028

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