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Theorem coafval 18014

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 6892	. . . . . 6 ⊢ (𝑐 = 𝐶 → (Arrow‘𝑐) = (Arrow‘𝐶))
3		coafval.a	. . . . . 6 ⊢ 𝐴 = (Arrow‘𝐶)
4	2, 3	eqtr4di 2791	. . . . 5 ⊢ (𝑐 = 𝐶 → (Arrow‘𝑐) = 𝐴)
5	4	rabeqdv 3448	. . . . 5 ⊢ (𝑐 = 𝐶 → {ℎ ∈ (Arrow‘𝑐) ∣ (cod_a‘ℎ) = (dom_a‘𝑔)} = {ℎ ∈ 𝐴 ∣ (cod_a‘ℎ) = (dom_a‘𝑔)})
6		fveq2 6892	. . . . . . . . 9 ⊢ (𝑐 = 𝐶 → (comp‘𝑐) = (comp‘𝐶))
7		coafval.x	. . . . . . . . 9 ⊢ ∙ = (comp‘𝐶)
8	6, 7	eqtr4di 2791	. . . . . . . 8 ⊢ (𝑐 = 𝐶 → (comp‘𝑐) = ∙ )
9	8	oveqd 7426	. . . . . . 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 7426	. . . . . 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 4888	. . . . 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 7484	. . . 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 18006	. . . 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 6906	. . . . 5 ⊢ 𝐴 ∈ V
15	14	rabex 5333	. . . . 5 ⊢ {ℎ ∈ 𝐴 ∣ (cod_a‘ℎ) = (dom_a‘𝑔)} ∈ V
16	14, 15	mpoex 8066	. . . 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 6999	. . 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 7029	. . . 4 ⊢ (¬ 𝐶 ∈ Cat → (comp_a‘𝐶) = ∅)
19	3	arwrcl 17994	. . . . . . . 8 ⊢ (𝑓 ∈ 𝐴 → 𝐶 ∈ Cat)
20	19	con3i 154	. . . . . . 7 ⊢ (¬ 𝐶 ∈ Cat → ¬ 𝑓 ∈ 𝐴)
21	20	eq0rdv 4405	. . . . . 6 ⊢ (¬ 𝐶 ∈ Cat → 𝐴 = ∅)
22		eqidd 2734	. . . . . 6 ⊢ (¬ 𝐶 ∈ Cat → {ℎ ∈ 𝐴 ∣ (cod_a‘ℎ) = (dom_a‘𝑔)} = {ℎ ∈ 𝐴 ∣ (cod_a‘ℎ) = (dom_a‘𝑔)})
23		eqidd 2734	. . . . . 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 7484	. . . . 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 7494	. . . . 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 2789	. . . 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 2776	. . 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 2761	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 1542 ∈ wcel 2107 {crab 3433 ∅c0 4323 ⟨cop 4635 ⟨cotp 4637 ‘cfv 6544 (class class class)co 7409 ∈ cmpo 7411 2^nd c2nd 7974 compcco 17209 Catccat 17608 dom_acdoma 17970 cod_accoda 17971 Arrowcarw 17972 comp_accoa 18004

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 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7725

This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-ral 3063 df-rex 3072 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-ot 4638 df-uni 4910 df-iun 5000 df-br 5150 df-opab 5212 df-mpt 5233 df-id 5575 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-ov 7412 df-oprab 7413 df-mpo 7414 df-1st 7975 df-2nd 7976 df-arw 17977 df-coa 18006

This theorem is referenced by: eldmcoa 18015 dmcoass 18016 coaval 18018 coapm 18021

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