Theorem coafval 17988

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

Hypotheses

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

Assertion

Ref	Expression
coafval	⊢ · = (𝑔 ∈ 𝐴, 𝑓 ∈ {ℎ ∈ 𝐴 ∣ (coda‘ℎ) = (doma‘𝑔)} ↦ ⟨(doma‘𝑓), (coda‘𝑔), ((2nd ‘𝑔)(⟨(doma‘𝑓), (doma‘𝑔)⟩ ∙ (coda‘𝑔))(2nd ‘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 ⊢ · = (compa‘𝐶)
2		fveq2 6834	. . . . . 6 ⊢ (𝑐 = 𝐶 → (Arrow‘𝑐) = (Arrow‘𝐶))
3		coafval.a	. . . . . 6 ⊢ 𝐴 = (Arrow‘𝐶)
4	2, 3	eqtr4di 2789	. . . . 5 ⊢ (𝑐 = 𝐶 → (Arrow‘𝑐) = 𝐴)
5	4	rabeqdv 3414	. . . . 5 ⊢ (𝑐 = 𝐶 → {ℎ ∈ (Arrow‘𝑐) ∣ (coda‘ℎ) = (doma‘𝑔)} = {ℎ ∈ 𝐴 ∣ (coda‘ℎ) = (doma‘𝑔)})
6		fveq2 6834	. . . . . . . . 9 ⊢ (𝑐 = 𝐶 → (comp‘𝑐) = (comp‘𝐶))
7		coafval.x	. . . . . . . . 9 ⊢ ∙ = (comp‘𝐶)
8	6, 7	eqtr4di 2789	. . . . . . . 8 ⊢ (𝑐 = 𝐶 → (comp‘𝑐) = ∙ )
9	8	oveqd 7375	. . . . . . 7 ⊢ (𝑐 = 𝐶 → (⟨(doma‘𝑓), (doma‘𝑔)⟩(comp‘𝑐)(coda‘𝑔)) = (⟨(doma‘𝑓), (doma‘𝑔)⟩ ∙ (coda‘𝑔)))
10	9	oveqd 7375	. . . . . 6 ⊢ (𝑐 = 𝐶 → ((2nd ‘𝑔)(⟨(doma‘𝑓), (doma‘𝑔)⟩(comp‘𝑐)(coda‘𝑔))(2nd ‘𝑓)) = ((2nd ‘𝑔)(⟨(doma‘𝑓), (doma‘𝑔)⟩ ∙ (coda‘𝑔))(2nd ‘𝑓)))
11	10	oteq3d 4843	. . . . 5 ⊢ (𝑐 = 𝐶 → ⟨(doma‘𝑓), (coda‘𝑔), ((2nd ‘𝑔)(⟨(doma‘𝑓), (doma‘𝑔)⟩(comp‘𝑐)(coda‘𝑔))(2nd ‘𝑓))⟩ = ⟨(doma‘𝑓), (coda‘𝑔), ((2nd ‘𝑔)(⟨(doma‘𝑓), (doma‘𝑔)⟩ ∙ (coda‘𝑔))(2nd ‘𝑓))⟩)
12	4, 5, 11	mpoeq123dv 7433	. . . 4 ⊢ (𝑐 = 𝐶 → (𝑔 ∈ (Arrow‘𝑐), 𝑓 ∈ {ℎ ∈ (Arrow‘𝑐) ∣ (coda‘ℎ) = (doma‘𝑔)} ↦ ⟨(doma‘𝑓), (coda‘𝑔), ((2nd ‘𝑔)(⟨(doma‘𝑓), (doma‘𝑔)⟩(comp‘𝑐)(coda‘𝑔))(2nd ‘𝑓))⟩) = (𝑔 ∈ 𝐴, 𝑓 ∈ {ℎ ∈ 𝐴 ∣ (coda‘ℎ) = (doma‘𝑔)} ↦ ⟨(doma‘𝑓), (coda‘𝑔), ((2nd ‘𝑔)(⟨(doma‘𝑓), (doma‘𝑔)⟩ ∙ (coda‘𝑔))(2nd ‘𝑓))⟩))
13		df-coa 17980	. . . 4 ⊢ compa = (𝑐 ∈ Cat ↦ (𝑔 ∈ (Arrow‘𝑐), 𝑓 ∈ {ℎ ∈ (Arrow‘𝑐) ∣ (coda‘ℎ) = (doma‘𝑔)} ↦ ⟨(doma‘𝑓), (coda‘𝑔), ((2nd ‘𝑔)(⟨(doma‘𝑓), (doma‘𝑔)⟩(comp‘𝑐)(coda‘𝑔))(2nd ‘𝑓))⟩))
14	3	fvexi 6848	. . . . 5 ⊢ 𝐴 ∈ V
15	14	rabex 5284	. . . . 5 ⊢ {ℎ ∈ 𝐴 ∣ (coda‘ℎ) = (doma‘𝑔)} ∈ V
16	14, 15	mpoex 8023	. . . 4 ⊢ (𝑔 ∈ 𝐴, 𝑓 ∈ {ℎ ∈ 𝐴 ∣ (coda‘ℎ) = (doma‘𝑔)} ↦ ⟨(doma‘𝑓), (coda‘𝑔), ((2nd ‘𝑔)(⟨(doma‘𝑓), (doma‘𝑔)⟩ ∙ (coda‘𝑔))(2nd ‘𝑓))⟩) ∈ V
17	12, 13, 16	fvmpt 6941	. . 3 ⊢ (𝐶 ∈ Cat → (compa‘𝐶) = (𝑔 ∈ 𝐴, 𝑓 ∈ {ℎ ∈ 𝐴 ∣ (coda‘ℎ) = (doma‘𝑔)} ↦ ⟨(doma‘𝑓), (coda‘𝑔), ((2nd ‘𝑔)(⟨(doma‘𝑓), (doma‘𝑔)⟩ ∙ (coda‘𝑔))(2nd ‘𝑓))⟩))
18	13	fvmptndm 6972	. . . 4 ⊢ (¬ 𝐶 ∈ Cat → (compa‘𝐶) = ∅)
19	3	arwrcl 17968	. . . . . . . 8 ⊢ (𝑓 ∈ 𝐴 → 𝐶 ∈ Cat)
20	19	con3i 154	. . . . . . 7 ⊢ (¬ 𝐶 ∈ Cat → ¬ 𝑓 ∈ 𝐴)
21	20	eq0rdv 4359	. . . . . 6 ⊢ (¬ 𝐶 ∈ Cat → 𝐴 = ∅)
22		eqidd 2737	. . . . . 6 ⊢ (¬ 𝐶 ∈ Cat → {ℎ ∈ 𝐴 ∣ (coda‘ℎ) = (doma‘𝑔)} = {ℎ ∈ 𝐴 ∣ (coda‘ℎ) = (doma‘𝑔)})
23		eqidd 2737	. . . . . 6 ⊢ (¬ 𝐶 ∈ Cat → ⟨(doma‘𝑓), (coda‘𝑔), ((2nd ‘𝑔)(⟨(doma‘𝑓), (doma‘𝑔)⟩ ∙ (coda‘𝑔))(2nd ‘𝑓))⟩ = ⟨(doma‘𝑓), (coda‘𝑔), ((2nd ‘𝑔)(⟨(doma‘𝑓), (doma‘𝑔)⟩ ∙ (coda‘𝑔))(2nd ‘𝑓))⟩)
24	21, 22, 23	mpoeq123dv 7433	. . . . 5 ⊢ (¬ 𝐶 ∈ Cat → (𝑔 ∈ 𝐴, 𝑓 ∈ {ℎ ∈ 𝐴 ∣ (coda‘ℎ) = (doma‘𝑔)} ↦ ⟨(doma‘𝑓), (coda‘𝑔), ((2nd ‘𝑔)(⟨(doma‘𝑓), (doma‘𝑔)⟩ ∙ (coda‘𝑔))(2nd ‘𝑓))⟩) = (𝑔 ∈ ∅, 𝑓 ∈ {ℎ ∈ 𝐴 ∣ (coda‘ℎ) = (doma‘𝑔)} ↦ ⟨(doma‘𝑓), (coda‘𝑔), ((2nd ‘𝑔)(⟨(doma‘𝑓), (doma‘𝑔)⟩ ∙ (coda‘𝑔))(2nd ‘𝑓))⟩))
25		mpo0 7443	. . . . 5 ⊢ (𝑔 ∈ ∅, 𝑓 ∈ {ℎ ∈ 𝐴 ∣ (coda‘ℎ) = (doma‘𝑔)} ↦ ⟨(doma‘𝑓), (coda‘𝑔), ((2nd ‘𝑔)(⟨(doma‘𝑓), (doma‘𝑔)⟩ ∙ (coda‘𝑔))(2nd ‘𝑓))⟩) = ∅
26	24, 25	eqtrdi 2787	. . . 4 ⊢ (¬ 𝐶 ∈ Cat → (𝑔 ∈ 𝐴, 𝑓 ∈ {ℎ ∈ 𝐴 ∣ (coda‘ℎ) = (doma‘𝑔)} ↦ ⟨(doma‘𝑓), (coda‘𝑔), ((2nd ‘𝑔)(⟨(doma‘𝑓), (doma‘𝑔)⟩ ∙ (coda‘𝑔))(2nd ‘𝑓))⟩) = ∅)
27	18, 26	eqtr4d 2774	. . 3 ⊢ (¬ 𝐶 ∈ Cat → (compa‘𝐶) = (𝑔 ∈ 𝐴, 𝑓 ∈ {ℎ ∈ 𝐴 ∣ (coda‘ℎ) = (doma‘𝑔)} ↦ ⟨(doma‘𝑓), (coda‘𝑔), ((2nd ‘𝑔)(⟨(doma‘𝑓), (doma‘𝑔)⟩ ∙ (coda‘𝑔))(2nd ‘𝑓))⟩))
28	17, 27	pm2.61i 182	. 2 ⊢ (compa‘𝐶) = (𝑔 ∈ 𝐴, 𝑓 ∈ {ℎ ∈ 𝐴 ∣ (coda‘ℎ) = (doma‘𝑔)} ↦ ⟨(doma‘𝑓), (coda‘𝑔), ((2nd ‘𝑔)(⟨(doma‘𝑓), (doma‘𝑔)⟩ ∙ (coda‘𝑔))(2nd ‘𝑓))⟩)
29	1, 28	eqtri 2759	1 ⊢ · = (𝑔 ∈ 𝐴, 𝑓 ∈ {ℎ ∈ 𝐴 ∣ (coda‘ℎ) = (doma‘𝑔)} ↦ ⟨(doma‘𝑓), (coda‘𝑔), ((2nd ‘𝑔)(⟨(doma‘𝑓), (doma‘𝑔)⟩ ∙ (coda‘𝑔))(2nd ‘𝑓))⟩)

Syntax hints:  ¬ wn 3   = wceq 1541   ∈ wcel 2113  {crab 3399  ∅c0 4285  ⟨cop 4586  ⟨cotp 4588  ‘cfv 6492  (class class class)co 7358   ∈ cmpo 7360  2^nd c2nd 7932  compcco 17189  Catccat 17587  dom_acdoma 17944  cod_accoda 17945  Arrowcarw 17946  comp_accoa 17978

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 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-rep 5224  ax-sep 5241  ax-nul 5251  ax-pow 5310  ax-pr 5377  ax-un 7680

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  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 3061  df-reu 3351  df-rab 3400  df-v 3442  df-sbc 3741  df-csb 3850  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4581  df-pr 4583  df-op 4587  df-ot 4589  df-uni 4864  df-iun 4948  df-br 5099  df-opab 5161  df-mpt 5180  df-id 5519  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-ov 7361  df-oprab 7362  df-mpo 7363  df-1st 7933  df-2nd 7934  df-arw 17951  df-coa 17980

This theorem is referenced by:  eldmcoa  17989  dmcoass  17990  coaval  17992  coapm  17995