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Theorem homacd 18008

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 17992	. . . 4 ⊢ cod_a = (2^nd ∘ 1^st )
2	1	fveq1i 6841	. . 3 ⊢ (cod_a‘𝐹) = ((2^nd ∘ 1^st )‘𝐹)
3		fo1st 7962	. . . . 5 ⊢ 1^st :V–onto→V
4		fof 6752	. . . . 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 6939	. . . 4 ⊢ ((1^st :V⟶V ∧ 𝐹 ∈ V) → ((2^nd ∘ 1^st )‘𝐹) = (2^nd ‘(1^st ‘𝐹)))
8	5, 6, 7	sylancr 588	. . 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 18003	. . . . 5 ⊢ Rel (𝑋𝐻𝑌)
12		1st2ndbr 7995	. . . . 5 ⊢ ((Rel (𝑋𝐻𝑌) ∧ 𝐹 ∈ (𝑋𝐻𝑌)) → (1^st ‘𝐹)(𝑋𝐻𝑌)(2^nd ‘𝐹))
13	11, 12	mpan 691	. . . 4 ⊢ (𝐹 ∈ (𝑋𝐻𝑌) → (1^st ‘𝐹)(𝑋𝐻𝑌)(2^nd ‘𝐹))
14	10	homa1 18004	. . . 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 6844	. 2 ⊢ (𝐹 ∈ (𝑋𝐻𝑌) → (2^nd ‘(1^st ‘𝐹)) = (2^nd ‘⟨𝑋, 𝑌⟩))
17		eqid 2736	. . . 4 ⊢ (Base‘𝐶) = (Base‘𝐶)
18	10, 17	homarcl2 18002	. . 3 ⊢ (𝐹 ∈ (𝑋𝐻𝑌) → (𝑋 ∈ (Base‘𝐶) ∧ 𝑌 ∈ (Base‘𝐶)))
19		op2ndg 7955	. . 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 1542 ∈ wcel 2114 Vcvv 3429 ⟨cop 4573 class class class wbr 5085 ∘ ccom 5635 Rel wrel 5636 ⟶wf 6494 –onto→wfo 6496 ‘cfv 6498 (class class class)co 7367 1^st c1st 7940 2^nd c2nd 7941 Basecbs 17179 cod_accoda 17988 Hom_achoma 17990

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2708 ax-rep 5212 ax-sep 5231 ax-nul 5241 ax-pow 5307 ax-pr 5375 ax-un 7689

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2539 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2811 df-nfc 2885 df-ne 2933 df-ral 3052 df-rex 3062 df-reu 3343 df-rab 3390 df-v 3431 df-sbc 3729 df-csb 3838 df-dif 3892 df-un 3894 df-in 3896 df-ss 3906 df-nul 4274 df-if 4467 df-pw 4543 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4851 df-iun 4935 df-br 5086 df-opab 5148 df-mpt 5167 df-id 5526 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-iota 6454 df-fun 6500 df-fn 6501 df-f 6502 df-f1 6503 df-fo 6504 df-f1o 6505 df-fv 6506 df-ov 7370 df-1st 7942 df-2nd 7943 df-coda 17992 df-homa 17993

This theorem is referenced by: arwhoma 18012 idacd 18029 homdmcoa 18034 coaval 18035

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