Theorem eldmcoa 18119

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 5149	. 2 ⊢ (𝐺dom · 𝐹 ↔ ⟨𝐺, 𝐹⟩ ∈ dom · )
2		otex 5476	. . . . . 6 ⊢ ⟨(doma‘𝑓), (coda‘𝑔), ((2nd ‘𝑔)(⟨(doma‘𝑓), (doma‘𝑔)⟩(comp‘𝐶)(coda‘𝑔))(2nd ‘𝑓))⟩ ∈ V
3	2	rgen2w 3064	. . . . 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 2735	. . . . . . 7 ⊢ (comp‘𝐶) = (comp‘𝐶)
7	4, 5, 6	coafval 18118	. . . . . 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 8091	. . . . 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 6748	. . 3 ⊢ dom · = ∪ 𝑔 ∈ 𝐴 ({𝑔} × {ℎ ∈ 𝐴 ∣ (coda‘ℎ) = (doma‘𝑔)})
11	10	eleq2i 2831	. 2 ⊢ (⟨𝐺, 𝐹⟩ ∈ dom · ↔ ⟨𝐺, 𝐹⟩ ∈ ∪ 𝑔 ∈ 𝐴 ({𝑔} × {ℎ ∈ 𝐴 ∣ (coda‘ℎ) = (doma‘𝑔)}))
12		fveq2 6907	. . . . . 6 ⊢ (𝑔 = 𝐺 → (doma‘𝑔) = (doma‘𝐺))
13	12	eqeq2d 2746	. . . . 5 ⊢ (𝑔 = 𝐺 → ((coda‘ℎ) = (doma‘𝑔) ↔ (coda‘ℎ) = (doma‘𝐺)))
14	13	rabbidv 3441	. . . 4 ⊢ (𝑔 = 𝐺 → {ℎ ∈ 𝐴 ∣ (coda‘ℎ) = (doma‘𝑔)} = {ℎ ∈ 𝐴 ∣ (coda‘ℎ) = (doma‘𝐺)})
15	14	opeliunxp2 5852	. . 3 ⊢ (⟨𝐺, 𝐹⟩ ∈ ∪ 𝑔 ∈ 𝐴 ({𝑔} × {ℎ ∈ 𝐴 ∣ (coda‘ℎ) = (doma‘𝑔)}) ↔ (𝐺 ∈ 𝐴 ∧ 𝐹 ∈ {ℎ ∈ 𝐴 ∣ (coda‘ℎ) = (doma‘𝐺)}))
16		fveqeq2 6916	. . . . 5 ⊢ (ℎ = 𝐹 → ((coda‘ℎ) = (doma‘𝐺) ↔ (coda‘𝐹) = (doma‘𝐺)))
17	16	elrab 3695	. . . 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 1094	. . . 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 1086   = wceq 1537   ∈ wcel 2106  ∀wral 3059  {crab 3433  Vcvv 3478  {csn 4631  ⟨cop 4637  ⟨cotp 4639  ∪ ciun 4996   class class class wbr 5148   × cxp 5687  dom cdm 5689  ⟶wf 6559  ‘cfv 6563  (class class class)co 7431  2^nd c2nd 8012  compcco 17310  dom_acdoma 18074  cod_accoda 18075  Arrowcarw 18076  comp_accoa 18108

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-ot 4640  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8013  df-2nd 8014  df-arw 18081  df-coa 18110

This theorem is referenced by:  homdmcoa  18121  coapm  18125