Theorem coafval 17326

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 6672	. . . . . 6 ⊢ (𝑐 = 𝐶 → (Arrow‘𝑐) = (Arrow‘𝐶))
3		coafval.a	. . . . . 6 ⊢ 𝐴 = (Arrow‘𝐶)
4	2, 3	syl6eqr 2876	. . . . 5 ⊢ (𝑐 = 𝐶 → (Arrow‘𝑐) = 𝐴)
5	4	rabeqdv 3486	. . . . 5 ⊢ (𝑐 = 𝐶 → {ℎ ∈ (Arrow‘𝑐) ∣ (coda‘ℎ) = (doma‘𝑔)} = {ℎ ∈ 𝐴 ∣ (coda‘ℎ) = (doma‘𝑔)})
6		fveq2 6672	. . . . . . . . 9 ⊢ (𝑐 = 𝐶 → (comp‘𝑐) = (comp‘𝐶))
7		coafval.x	. . . . . . . . 9 ⊢ ∙ = (comp‘𝐶)
8	6, 7	syl6eqr 2876	. . . . . . . 8 ⊢ (𝑐 = 𝐶 → (comp‘𝑐) = ∙ )
9	8	oveqd 7175	. . . . . . 7 ⊢ (𝑐 = 𝐶 → (⟨(doma‘𝑓), (doma‘𝑔)⟩(comp‘𝑐)(coda‘𝑔)) = (⟨(doma‘𝑓), (doma‘𝑔)⟩ ∙ (coda‘𝑔)))
10	9	oveqd 7175	. . . . . 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 4819	. . . . 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 7231	. . . 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 17318	. . . 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 6686	. . . . 5 ⊢ 𝐴 ∈ V
15	14	rabex 5237	. . . . 5 ⊢ {ℎ ∈ 𝐴 ∣ (coda‘ℎ) = (doma‘𝑔)} ∈ V
16	14, 15	mpoex 7779	. . . 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 6770	. . 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 6800	. . . 4 ⊢ (¬ 𝐶 ∈ Cat → (compa‘𝐶) = ∅)
19	3	arwrcl 17306	. . . . . . . 8 ⊢ (𝑓 ∈ 𝐴 → 𝐶 ∈ Cat)
20	19	con3i 157	. . . . . . 7 ⊢ (¬ 𝐶 ∈ Cat → ¬ 𝑓 ∈ 𝐴)
21	20	eq0rdv 4359	. . . . . 6 ⊢ (¬ 𝐶 ∈ Cat → 𝐴 = ∅)
22		eqidd 2824	. . . . . 6 ⊢ (¬ 𝐶 ∈ Cat → {ℎ ∈ 𝐴 ∣ (coda‘ℎ) = (doma‘𝑔)} = {ℎ ∈ 𝐴 ∣ (coda‘ℎ) = (doma‘𝑔)})
23		eqidd 2824	. . . . . 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 7231	. . . . 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 7241	. . . . 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	syl6eq 2874	. . . 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 2861	. . 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 184	. 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 2846	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 1537   ∈ wcel 2114  {crab 3144  ∅c0 4293  ⟨cop 4575  ⟨cotp 4577  ‘cfv 6357  (class class class)co 7158   ∈ cmpo 7160  2^nd c2nd 7690  compcco 16579  Catccat 16937  dom_acdoma 17282  cod_accoda 17283  Arrowcarw 17284  comp_accoa 17316

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 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795  ax-rep 5192  ax-sep 5205  ax-nul 5212  ax-pow 5268  ax-pr 5332  ax-un 7463

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ne 3019  df-ral 3145  df-rex 3146  df-reu 3147  df-rab 3149  df-v 3498  df-sbc 3775  df-csb 3886  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-nul 4294  df-if 4470  df-pw 4543  df-sn 4570  df-pr 4572  df-op 4576  df-ot 4578  df-uni 4841  df-iun 4923  df-br 5069  df-opab 5131  df-mpt 5149  df-id 5462  df-xp 5563  df-rel 5564  df-cnv 5565  df-co 5566  df-dm 5567  df-rn 5568  df-res 5569  df-ima 5570  df-iota 6316  df-fun 6359  df-fn 6360  df-f 6361  df-f1 6362  df-fo 6363  df-f1o 6364  df-fv 6365  df-ov 7161  df-oprab 7162  df-mpo 7163  df-1st 7691  df-2nd 7692  df-arw 17289  df-coa 17318

This theorem is referenced by:  eldmcoa  17327  dmcoass  17328  coaval  17330  coapm  17333