Theorem eldmcoa 18110

Description: A pair ⟨𝐺, 𝐹⟩ is in the domain of the arrow composition, if the domain of 𝐺 equals the codomain of 𝐹. (In this case we say 𝐺 and 𝐹 are composable.) (Contributed by Mario Carneiro, 11-Jan-2017.)

Hypotheses

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

Assertion

Ref	Expression
eldmcoa	⊢ (𝐺dom · 𝐹 ↔ (𝐹 ∈ 𝐴 ∧ 𝐺 ∈ 𝐴 ∧ (coda‘𝐹) = (doma‘𝐺)))

Proof of Theorem eldmcoa

Dummy variables 𝑓 𝑔 ℎ are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-br 5144	. 2 ⊢ (𝐺dom · 𝐹 ↔ ⟨𝐺, 𝐹⟩ ∈ dom · )
2		otex 5470	. . . . . 6 ⊢ ⟨(doma‘𝑓), (coda‘𝑔), ((2nd ‘𝑔)(⟨(doma‘𝑓), (doma‘𝑔)⟩(comp‘𝐶)(coda‘𝑔))(2nd ‘𝑓))⟩ ∈ V
3	2	rgen2w 3066	. . . . 5 ⊢ ∀𝑔 ∈ 𝐴 ∀𝑓 ∈ {ℎ ∈ 𝐴 ∣ (coda‘ℎ) = (doma‘𝑔)}⟨(doma‘𝑓), (coda‘𝑔), ((2nd ‘𝑔)(⟨(doma‘𝑓), (doma‘𝑔)⟩(comp‘𝐶)(coda‘𝑔))(2nd ‘𝑓))⟩ ∈ V
4		coafval.o	. . . . . . 7 ⊢ · = (compa‘𝐶)
5		coafval.a	. . . . . . 7 ⊢ 𝐴 = (Arrow‘𝐶)
6		eqid 2737	. . . . . . 7 ⊢ (comp‘𝐶) = (comp‘𝐶)
7	4, 5, 6	coafval 18109	. . . . . 6 ⊢ · = (𝑔 ∈ 𝐴, 𝑓 ∈ {ℎ ∈ 𝐴 ∣ (coda‘ℎ) = (doma‘𝑔)} ↦ ⟨(doma‘𝑓), (coda‘𝑔), ((2nd ‘𝑔)(⟨(doma‘𝑓), (doma‘𝑔)⟩(comp‘𝐶)(coda‘𝑔))(2nd ‘𝑓))⟩)
8	7	fmpox 8092	. . . . 5 ⊢ (∀𝑔 ∈ 𝐴 ∀𝑓 ∈ {ℎ ∈ 𝐴 ∣ (coda‘ℎ) = (doma‘𝑔)}⟨(doma‘𝑓), (coda‘𝑔), ((2nd ‘𝑔)(⟨(doma‘𝑓), (doma‘𝑔)⟩(comp‘𝐶)(coda‘𝑔))(2nd ‘𝑓))⟩ ∈ V ↔ · :∪ 𝑔 ∈ 𝐴 ({𝑔} × {ℎ ∈ 𝐴 ∣ (coda‘ℎ) = (doma‘𝑔)})⟶V)
9	3, 8	mpbi 230	. . . 4 ⊢ · :∪ 𝑔 ∈ 𝐴 ({𝑔} × {ℎ ∈ 𝐴 ∣ (coda‘ℎ) = (doma‘𝑔)})⟶V
10	9	fdmi 6747	. . 3 ⊢ dom · = ∪ 𝑔 ∈ 𝐴 ({𝑔} × {ℎ ∈ 𝐴 ∣ (coda‘ℎ) = (doma‘𝑔)})
11	10	eleq2i 2833	. 2 ⊢ (⟨𝐺, 𝐹⟩ ∈ dom · ↔ ⟨𝐺, 𝐹⟩ ∈ ∪ 𝑔 ∈ 𝐴 ({𝑔} × {ℎ ∈ 𝐴 ∣ (coda‘ℎ) = (doma‘𝑔)}))
12		fveq2 6906	. . . . . 6 ⊢ (𝑔 = 𝐺 → (doma‘𝑔) = (doma‘𝐺))
13	12	eqeq2d 2748	. . . . 5 ⊢ (𝑔 = 𝐺 → ((coda‘ℎ) = (doma‘𝑔) ↔ (coda‘ℎ) = (doma‘𝐺)))
14	13	rabbidv 3444	. . . 4 ⊢ (𝑔 = 𝐺 → {ℎ ∈ 𝐴 ∣ (coda‘ℎ) = (doma‘𝑔)} = {ℎ ∈ 𝐴 ∣ (coda‘ℎ) = (doma‘𝐺)})
15	14	opeliunxp2 5849	. . 3 ⊢ (⟨𝐺, 𝐹⟩ ∈ ∪ 𝑔 ∈ 𝐴 ({𝑔} × {ℎ ∈ 𝐴 ∣ (coda‘ℎ) = (doma‘𝑔)}) ↔ (𝐺 ∈ 𝐴 ∧ 𝐹 ∈ {ℎ ∈ 𝐴 ∣ (coda‘ℎ) = (doma‘𝐺)}))
16		fveqeq2 6915	. . . . 5 ⊢ (ℎ = 𝐹 → ((coda‘ℎ) = (doma‘𝐺) ↔ (coda‘𝐹) = (doma‘𝐺)))
17	16	elrab 3692	. . . 4 ⊢ (𝐹 ∈ {ℎ ∈ 𝐴 ∣ (coda‘ℎ) = (doma‘𝐺)} ↔ (𝐹 ∈ 𝐴 ∧ (coda‘𝐹) = (doma‘𝐺)))
18	17	anbi2i 623	. . 3 ⊢ ((𝐺 ∈ 𝐴 ∧ 𝐹 ∈ {ℎ ∈ 𝐴 ∣ (coda‘ℎ) = (doma‘𝐺)}) ↔ (𝐺 ∈ 𝐴 ∧ (𝐹 ∈ 𝐴 ∧ (coda‘𝐹) = (doma‘𝐺))))
19		an12 645	. . . 4 ⊢ ((𝐺 ∈ 𝐴 ∧ (𝐹 ∈ 𝐴 ∧ (coda‘𝐹) = (doma‘𝐺))) ↔ (𝐹 ∈ 𝐴 ∧ (𝐺 ∈ 𝐴 ∧ (coda‘𝐹) = (doma‘𝐺))))
20		3anass 1095	. . . 4 ⊢ ((𝐹 ∈ 𝐴 ∧ 𝐺 ∈ 𝐴 ∧ (coda‘𝐹) = (doma‘𝐺)) ↔ (𝐹 ∈ 𝐴 ∧ (𝐺 ∈ 𝐴 ∧ (coda‘𝐹) = (doma‘𝐺))))
21	19, 20	bitr4i 278	. . 3 ⊢ ((𝐺 ∈ 𝐴 ∧ (𝐹 ∈ 𝐴 ∧ (coda‘𝐹) = (doma‘𝐺))) ↔ (𝐹 ∈ 𝐴 ∧ 𝐺 ∈ 𝐴 ∧ (coda‘𝐹) = (doma‘𝐺)))
22	15, 18, 21	3bitri 297	. 2 ⊢ (⟨𝐺, 𝐹⟩ ∈ ∪ 𝑔 ∈ 𝐴 ({𝑔} × {ℎ ∈ 𝐴 ∣ (coda‘ℎ) = (doma‘𝑔)}) ↔ (𝐹 ∈ 𝐴 ∧ 𝐺 ∈ 𝐴 ∧ (coda‘𝐹) = (doma‘𝐺)))
23	1, 11, 22	3bitri 297	1 ⊢ (𝐺dom · 𝐹 ↔ (𝐹 ∈ 𝐴 ∧ 𝐺 ∈ 𝐴 ∧ (coda‘𝐹) = (doma‘𝐺)))

Syntax hints:   ↔ wb 206   ∧ wa 395   ∧ w3a 1087   = wceq 1540   ∈ wcel 2108  ∀wral 3061  {crab 3436  Vcvv 3480  {csn 4626  ⟨cop 4632  ⟨cotp 4634  ∪ ciun 4991   class class class wbr 5143   × cxp 5683  dom cdm 5685  ⟶wf 6557  ‘cfv 6561  (class class class)co 7431  2^nd c2nd 8013  compcco 17309  dom_acdoma 18065  cod_accoda 18066  Arrowcarw 18067  comp_accoa 18099

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-ot 4635  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8014  df-2nd 8015  df-arw 18072  df-coa 18101

This theorem is referenced by:  homdmcoa  18112  coapm  18116