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Theorem coafval 18016

Description: The value of the composition of arrows. (Contributed by Mario Carneiro, 11-Jan-2017.)

**Hypotheses**
Ref	Expression
coafval.o	⊢ · = (comp_a‘𝐶)
coafval.a	⊢ 𝐴 = (Arrow‘𝐶)
coafval.x	⊢ ∙ = (comp‘𝐶)

**Assertion**
Ref	Expression
coafval	⊢ · = (𝑔 ∈ 𝐴, 𝑓 ∈ {ℎ ∈ 𝐴 ∣ (cod_a‘ℎ) = (dom_a‘𝑔)} ↦ ⟨(dom_a‘𝑓), (cod_a‘𝑔), ((2^nd ‘𝑔)(⟨(dom_a‘𝑓), (dom_a‘𝑔)⟩ ∙ (cod_a‘𝑔))(2^nd ‘𝑓))⟩)

Distinct variable groups: 𝑓,𝑔,ℎ,𝐴 𝐶,𝑓,𝑔,ℎ Allowed substitution hints: ∙ (𝑓,𝑔,ℎ) · (𝑓,𝑔,ℎ)

Proof of Theorem coafval Dummy variable 𝑐 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		coafval.o	. 2 ⊢ · = (comp_a‘𝐶)
2		fveq2 6891	. . . . . 6 ⊢ (𝑐 = 𝐶 → (Arrow‘𝑐) = (Arrow‘𝐶))
3		coafval.a	. . . . . 6 ⊢ 𝐴 = (Arrow‘𝐶)
4	2, 3	eqtr4di 2790	. . . . 5 ⊢ (𝑐 = 𝐶 → (Arrow‘𝑐) = 𝐴)
5	4	rabeqdv 3447	. . . . 5 ⊢ (𝑐 = 𝐶 → {ℎ ∈ (Arrow‘𝑐) ∣ (cod_a‘ℎ) = (dom_a‘𝑔)} = {ℎ ∈ 𝐴 ∣ (cod_a‘ℎ) = (dom_a‘𝑔)})
6		fveq2 6891	. . . . . . . . 9 ⊢ (𝑐 = 𝐶 → (comp‘𝑐) = (comp‘𝐶))
7		coafval.x	. . . . . . . . 9 ⊢ ∙ = (comp‘𝐶)
8	6, 7	eqtr4di 2790	. . . . . . . 8 ⊢ (𝑐 = 𝐶 → (comp‘𝑐) = ∙ )
9	8	oveqd 7428	. . . . . . 7 ⊢ (𝑐 = 𝐶 → (⟨(dom_a‘𝑓), (dom_a‘𝑔)⟩(comp‘𝑐)(cod_a‘𝑔)) = (⟨(dom_a‘𝑓), (dom_a‘𝑔)⟩ ∙ (cod_a‘𝑔)))
10	9	oveqd 7428	. . . . . 6 ⊢ (𝑐 = 𝐶 → ((2^nd ‘𝑔)(⟨(dom_a‘𝑓), (dom_a‘𝑔)⟩(comp‘𝑐)(cod_a‘𝑔))(2^nd ‘𝑓)) = ((2^nd ‘𝑔)(⟨(dom_a‘𝑓), (dom_a‘𝑔)⟩ ∙ (cod_a‘𝑔))(2^nd ‘𝑓)))
11	10	oteq3d 4887	. . . . 5 ⊢ (𝑐 = 𝐶 → ⟨(dom_a‘𝑓), (cod_a‘𝑔), ((2^nd ‘𝑔)(⟨(dom_a‘𝑓), (dom_a‘𝑔)⟩(comp‘𝑐)(cod_a‘𝑔))(2^nd ‘𝑓))⟩ = ⟨(dom_a‘𝑓), (cod_a‘𝑔), ((2^nd ‘𝑔)(⟨(dom_a‘𝑓), (dom_a‘𝑔)⟩ ∙ (cod_a‘𝑔))(2^nd ‘𝑓))⟩)
12	4, 5, 11	mpoeq123dv 7486	. . . 4 ⊢ (𝑐 = 𝐶 → (𝑔 ∈ (Arrow‘𝑐), 𝑓 ∈ {ℎ ∈ (Arrow‘𝑐) ∣ (cod_a‘ℎ) = (dom_a‘𝑔)} ↦ ⟨(dom_a‘𝑓), (cod_a‘𝑔), ((2^nd ‘𝑔)(⟨(dom_a‘𝑓), (dom_a‘𝑔)⟩(comp‘𝑐)(cod_a‘𝑔))(2^nd ‘𝑓))⟩) = (𝑔 ∈ 𝐴, 𝑓 ∈ {ℎ ∈ 𝐴 ∣ (cod_a‘ℎ) = (dom_a‘𝑔)} ↦ ⟨(dom_a‘𝑓), (cod_a‘𝑔), ((2^nd ‘𝑔)(⟨(dom_a‘𝑓), (dom_a‘𝑔)⟩ ∙ (cod_a‘𝑔))(2^nd ‘𝑓))⟩))
13		df-coa 18008	. . . 4 ⊢ comp_a = (𝑐 ∈ Cat ↦ (𝑔 ∈ (Arrow‘𝑐), 𝑓 ∈ {ℎ ∈ (Arrow‘𝑐) ∣ (cod_a‘ℎ) = (dom_a‘𝑔)} ↦ ⟨(dom_a‘𝑓), (cod_a‘𝑔), ((2^nd ‘𝑔)(⟨(dom_a‘𝑓), (dom_a‘𝑔)⟩(comp‘𝑐)(cod_a‘𝑔))(2^nd ‘𝑓))⟩))
14	3	fvexi 6905	. . . . 5 ⊢ 𝐴 ∈ V
15	14	rabex 5332	. . . . 5 ⊢ {ℎ ∈ 𝐴 ∣ (cod_a‘ℎ) = (dom_a‘𝑔)} ∈ V
16	14, 15	mpoex 8068	. . . 4 ⊢ (𝑔 ∈ 𝐴, 𝑓 ∈ {ℎ ∈ 𝐴 ∣ (cod_a‘ℎ) = (dom_a‘𝑔)} ↦ ⟨(dom_a‘𝑓), (cod_a‘𝑔), ((2^nd ‘𝑔)(⟨(dom_a‘𝑓), (dom_a‘𝑔)⟩ ∙ (cod_a‘𝑔))(2^nd ‘𝑓))⟩) ∈ V
17	12, 13, 16	fvmpt 6998	. . 3 ⊢ (𝐶 ∈ Cat → (comp_a‘𝐶) = (𝑔 ∈ 𝐴, 𝑓 ∈ {ℎ ∈ 𝐴 ∣ (cod_a‘ℎ) = (dom_a‘𝑔)} ↦ ⟨(dom_a‘𝑓), (cod_a‘𝑔), ((2^nd ‘𝑔)(⟨(dom_a‘𝑓), (dom_a‘𝑔)⟩ ∙ (cod_a‘𝑔))(2^nd ‘𝑓))⟩))
18	13	fvmptndm 7028	. . . 4 ⊢ (¬ 𝐶 ∈ Cat → (comp_a‘𝐶) = ∅)
19	3	arwrcl 17996	. . . . . . . 8 ⊢ (𝑓 ∈ 𝐴 → 𝐶 ∈ Cat)
20	19	con3i 154	. . . . . . 7 ⊢ (¬ 𝐶 ∈ Cat → ¬ 𝑓 ∈ 𝐴)
21	20	eq0rdv 4404	. . . . . 6 ⊢ (¬ 𝐶 ∈ Cat → 𝐴 = ∅)
22		eqidd 2733	. . . . . 6 ⊢ (¬ 𝐶 ∈ Cat → {ℎ ∈ 𝐴 ∣ (cod_a‘ℎ) = (dom_a‘𝑔)} = {ℎ ∈ 𝐴 ∣ (cod_a‘ℎ) = (dom_a‘𝑔)})
23		eqidd 2733	. . . . . 6 ⊢ (¬ 𝐶 ∈ Cat → ⟨(dom_a‘𝑓), (cod_a‘𝑔), ((2^nd ‘𝑔)(⟨(dom_a‘𝑓), (dom_a‘𝑔)⟩ ∙ (cod_a‘𝑔))(2^nd ‘𝑓))⟩ = ⟨(dom_a‘𝑓), (cod_a‘𝑔), ((2^nd ‘𝑔)(⟨(dom_a‘𝑓), (dom_a‘𝑔)⟩ ∙ (cod_a‘𝑔))(2^nd ‘𝑓))⟩)
24	21, 22, 23	mpoeq123dv 7486	. . . . 5 ⊢ (¬ 𝐶 ∈ Cat → (𝑔 ∈ 𝐴, 𝑓 ∈ {ℎ ∈ 𝐴 ∣ (cod_a‘ℎ) = (dom_a‘𝑔)} ↦ ⟨(dom_a‘𝑓), (cod_a‘𝑔), ((2^nd ‘𝑔)(⟨(dom_a‘𝑓), (dom_a‘𝑔)⟩ ∙ (cod_a‘𝑔))(2^nd ‘𝑓))⟩) = (𝑔 ∈ ∅, 𝑓 ∈ {ℎ ∈ 𝐴 ∣ (cod_a‘ℎ) = (dom_a‘𝑔)} ↦ ⟨(dom_a‘𝑓), (cod_a‘𝑔), ((2^nd ‘𝑔)(⟨(dom_a‘𝑓), (dom_a‘𝑔)⟩ ∙ (cod_a‘𝑔))(2^nd ‘𝑓))⟩))
25		mpo0 7496	. . . . 5 ⊢ (𝑔 ∈ ∅, 𝑓 ∈ {ℎ ∈ 𝐴 ∣ (cod_a‘ℎ) = (dom_a‘𝑔)} ↦ ⟨(dom_a‘𝑓), (cod_a‘𝑔), ((2^nd ‘𝑔)(⟨(dom_a‘𝑓), (dom_a‘𝑔)⟩ ∙ (cod_a‘𝑔))(2^nd ‘𝑓))⟩) = ∅
26	24, 25	eqtrdi 2788	. . . 4 ⊢ (¬ 𝐶 ∈ Cat → (𝑔 ∈ 𝐴, 𝑓 ∈ {ℎ ∈ 𝐴 ∣ (cod_a‘ℎ) = (dom_a‘𝑔)} ↦ ⟨(dom_a‘𝑓), (cod_a‘𝑔), ((2^nd ‘𝑔)(⟨(dom_a‘𝑓), (dom_a‘𝑔)⟩ ∙ (cod_a‘𝑔))(2^nd ‘𝑓))⟩) = ∅)
27	18, 26	eqtr4d 2775	. . 3 ⊢ (¬ 𝐶 ∈ Cat → (comp_a‘𝐶) = (𝑔 ∈ 𝐴, 𝑓 ∈ {ℎ ∈ 𝐴 ∣ (cod_a‘ℎ) = (dom_a‘𝑔)} ↦ ⟨(dom_a‘𝑓), (cod_a‘𝑔), ((2^nd ‘𝑔)(⟨(dom_a‘𝑓), (dom_a‘𝑔)⟩ ∙ (cod_a‘𝑔))(2^nd ‘𝑓))⟩))
28	17, 27	pm2.61i 182	. 2 ⊢ (comp_a‘𝐶) = (𝑔 ∈ 𝐴, 𝑓 ∈ {ℎ ∈ 𝐴 ∣ (cod_a‘ℎ) = (dom_a‘𝑔)} ↦ ⟨(dom_a‘𝑓), (cod_a‘𝑔), ((2^nd ‘𝑔)(⟨(dom_a‘𝑓), (dom_a‘𝑔)⟩ ∙ (cod_a‘𝑔))(2^nd ‘𝑓))⟩)
29	1, 28	eqtri 2760	1 ⊢ · = (𝑔 ∈ 𝐴, 𝑓 ∈ {ℎ ∈ 𝐴 ∣ (cod_a‘ℎ) = (dom_a‘𝑔)} ↦ ⟨(dom_a‘𝑓), (cod_a‘𝑔), ((2^nd ‘𝑔)(⟨(dom_a‘𝑓), (dom_a‘𝑔)⟩ ∙ (cod_a‘𝑔))(2^nd ‘𝑓))⟩)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 = wceq 1541 ∈ wcel 2106 {crab 3432 ∅c0 4322 ⟨cop 4634 ⟨cotp 4636 ‘cfv 6543 (class class class)co 7411 ∈ cmpo 7413 2^nd c2nd 7976 compcco 17211 Catccat 17610 dom_acdoma 17972 cod_accoda 17973 Arrowcarw 17974 comp_accoa 18006

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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7727

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3377 df-rab 3433 df-v 3476 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-ot 4637 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 7414 df-oprab 7415 df-mpo 7416 df-1st 7977 df-2nd 7978 df-arw 17979 df-coa 18008

This theorem is referenced by: eldmcoa 18017 dmcoass 18018 coaval 18020 coapm 18023

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