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Theorem homacd 17756

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 17740	. . . 4 ⊢ cod_a = (2^nd ∘ 1^st )
2	1	fveq1i 6775	. . 3 ⊢ (cod_a‘𝐹) = ((2^nd ∘ 1^st )‘𝐹)
3		fo1st 7851	. . . . 5 ⊢ 1^st :V–onto→V
4		fof 6688	. . . . 5 ⊢ (1^st :V–onto→V → 1^st :V⟶V)
5	3, 4	ax-mp 5	. . . 4 ⊢ 1^st :V⟶V
6		elex 3450	. . . 4 ⊢ (𝐹 ∈ (𝑋𝐻𝑌) → 𝐹 ∈ V)
7		fvco3 6867	. . . 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 2790	. 2 ⊢ (𝐹 ∈ (𝑋𝐻𝑌) → (cod_a‘𝐹) = (2^nd ‘(1^st ‘𝐹)))
10		homahom.h	. . . . . 6 ⊢ 𝐻 = (Hom_a‘𝐶)
11	10	homarel 17751	. . . . 5 ⊢ Rel (𝑋𝐻𝑌)
12		1st2ndbr 7883	. . . . 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 17752	. . . 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 6778	. 2 ⊢ (𝐹 ∈ (𝑋𝐻𝑌) → (2^nd ‘(1^st ‘𝐹)) = (2^nd ‘⟨𝑋, 𝑌⟩))
17		eqid 2738	. . . 4 ⊢ (Base‘𝐶) = (Base‘𝐶)
18	10, 17	homarcl2 17750	. . 3 ⊢ (𝐹 ∈ (𝑋𝐻𝑌) → (𝑋 ∈ (Base‘𝐶) ∧ 𝑌 ∈ (Base‘𝐶)))
19		op2ndg 7844	. . 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 2782	1 ⊢ (𝐹 ∈ (𝑋𝐻𝑌) → (cod_a‘𝐹) = 𝑌)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 396 = wceq 1539 ∈ wcel 2106 Vcvv 3432 ⟨cop 4567 class class class wbr 5074 ∘ ccom 5593 Rel wrel 5594 ⟶wf 6429 –onto→wfo 6431 ‘cfv 6433 (class class class)co 7275 1^st c1st 7829 2^nd c2nd 7830 Basecbs 16912 cod_accoda 17736 Hom_achoma 17738

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-rep 5209 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7588

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-ral 3069 df-rex 3070 df-reu 3072 df-rab 3073 df-v 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-iun 4926 df-br 5075 df-opab 5137 df-mpt 5158 df-id 5489 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-f1 6438 df-fo 6439 df-f1o 6440 df-fv 6441 df-ov 7278 df-1st 7831 df-2nd 7832 df-coda 17740 df-homa 17741

This theorem is referenced by: arwhoma 17760 idacd 17777 homdmcoa 17782 coaval 17783

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