Theorem coafval 18088

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 6862	. . . . . 6 ⊢ (𝑐 = 𝐶 → (Arrow‘𝑐) = (Arrow‘𝐶))
3		coafval.a	. . . . . 6 ⊢ 𝐴 = (Arrow‘𝐶)
4	2, 3	eqtr4di 2814	. . . . 5 ⊢ (𝑐 = 𝐶 → (Arrow‘𝑐) = 𝐴)
5	4	rabeqdv 3428	. . . . 5 ⊢ (𝑐 = 𝐶 → {ℎ ∈ (Arrow‘𝑐) ∣ (coda‘ℎ) = (doma‘𝑔)} = {ℎ ∈ 𝐴 ∣ (coda‘ℎ) = (doma‘𝑔)})
6		fveq2 6862	. . . . . . . . 9 ⊢ (𝑐 = 𝐶 → (comp‘𝑐) = (comp‘𝐶))
7		coafval.x	. . . . . . . . 9 ⊢ ∙ = (comp‘𝐶)
8	6, 7	eqtr4di 2814	. . . . . . . 8 ⊢ (𝑐 = 𝐶 → (comp‘𝑐) = ∙ )
9	8	oveqd 7408	. . . . . . 7 ⊢ (𝑐 = 𝐶 → (⟨(doma‘𝑓), (doma‘𝑔)⟩(comp‘𝑐)(coda‘𝑔)) = (⟨(doma‘𝑓), (doma‘𝑔)⟩ ∙ (coda‘𝑔)))
10	9	oveqd 7408	. . . . . 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 4842	. . . . 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 7466	. . . 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 18080	. . . 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 6876	. . . . 5 ⊢ 𝐴 ∈ V
15	14	rabex 5292	. . . . 5 ⊢ {ℎ ∈ 𝐴 ∣ (coda‘ℎ) = (doma‘𝑔)} ∈ V
16	14, 15	mpoex 8055	. . . 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 6970	. . 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 7002	. . . 4 ⊢ (¬ 𝐶 ∈ Cat → (compa‘𝐶) = ∅)
19	3	arwrcl 18068	. . . . . . . 8 ⊢ (𝑓 ∈ 𝐴 → 𝐶 ∈ Cat)
20	19	con3i 154	. . . . . . 7 ⊢ (¬ 𝐶 ∈ Cat → ¬ 𝑓 ∈ 𝐴)
21	20	eq0rdv 4358	. . . . . 6 ⊢ (¬ 𝐶 ∈ Cat → 𝐴 = ∅)
22		eqidd 2762	. . . . . 6 ⊢ (¬ 𝐶 ∈ Cat → {ℎ ∈ 𝐴 ∣ (coda‘ℎ) = (doma‘𝑔)} = {ℎ ∈ 𝐴 ∣ (coda‘ℎ) = (doma‘𝑔)})
23		eqidd 2762	. . . . . 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 7466	. . . . 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 7476	. . . . 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 2812	. . . 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 2799	. . 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 183	. 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 2784	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 1559   ∈ wcel 2141  {crab 3413  ∅c0 4283  ⟨cop 4585  ⟨cotp 4587  ‘cfv 6516  (class class class)co 7391   ∈ cmpo 7393  2^nd c2nd 7964  compcco 17289  Catccat 17687  dom_acdoma 18044  cod_accoda 18045  Arrowcarw 18046  comp_accoa 18078

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-rep 5224  ax-sep 5243  ax-nul 5253  ax-pow 5319  ax-pr 5387  ax-un 7713

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3076  df-rex 3086  df-reu 3367  df-rab 3414  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4284  df-if 4478  df-pw 4554  df-sn 4580  df-pr 4582  df-op 4586  df-ot 4588  df-uni 4863  df-iun 4948  df-br 5098  df-opab 5160  df-mpt 5179  df-id 5538  df-xp 5649  df-rel 5650  df-cnv 5651  df-co 5652  df-dm 5653  df-rn 5654  df-res 5655  df-ima 5656  df-iota 6472  df-fun 6518  df-fn 6519  df-f 6520  df-f1 6521  df-fo 6522  df-f1o 6523  df-fv 6524  df-ov 7394  df-oprab 7395  df-mpo 7396  df-1st 7965  df-2nd 7966  df-arw 18051  df-coa 18080

This theorem is referenced by:  eldmcoa  18089  dmcoass  18090  coaval  18092  coapm  18095