Theorem eldmcoa 18027

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 5087	. 2 ⊢ (𝐺dom · 𝐹 ↔ ⟨𝐺, 𝐹⟩ ∈ dom · )
2		otex 5415	. . . . . 6 ⊢ ⟨(doma‘𝑓), (coda‘𝑔), ((2nd ‘𝑔)(⟨(doma‘𝑓), (doma‘𝑔)⟩(comp‘𝐶)(coda‘𝑔))(2nd ‘𝑓))⟩ ∈ V
3	2	rgen2w 3057	. . . . 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 18026	. . . . . 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 8015	. . . . 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 6675	. . 3 ⊢ dom · = ∪ 𝑔 ∈ 𝐴 ({𝑔} × {ℎ ∈ 𝐴 ∣ (coda‘ℎ) = (doma‘𝑔)})
11	10	eleq2i 2829	. 2 ⊢ (⟨𝐺, 𝐹⟩ ∈ dom · ↔ ⟨𝐺, 𝐹⟩ ∈ ∪ 𝑔 ∈ 𝐴 ({𝑔} × {ℎ ∈ 𝐴 ∣ (coda‘ℎ) = (doma‘𝑔)}))
12		fveq2 6836	. . . . . 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 3397	. . . 4 ⊢ (𝑔 = 𝐺 → {ℎ ∈ 𝐴 ∣ (coda‘ℎ) = (doma‘𝑔)} = {ℎ ∈ 𝐴 ∣ (coda‘ℎ) = (doma‘𝐺)})
15	14	opeliunxp2 5789	. . 3 ⊢ (⟨𝐺, 𝐹⟩ ∈ ∪ 𝑔 ∈ 𝐴 ({𝑔} × {ℎ ∈ 𝐴 ∣ (coda‘ℎ) = (doma‘𝑔)}) ↔ (𝐺 ∈ 𝐴 ∧ 𝐹 ∈ {ℎ ∈ 𝐴 ∣ (coda‘ℎ) = (doma‘𝐺)}))
16		fveqeq2 6845	. . . . 5 ⊢ (ℎ = 𝐹 → ((coda‘ℎ) = (doma‘𝐺) ↔ (coda‘𝐹) = (doma‘𝐺)))
17	16	elrab 3635	. . . 4 ⊢ (𝐹 ∈ {ℎ ∈ 𝐴 ∣ (coda‘ℎ) = (doma‘𝐺)} ↔ (𝐹 ∈ 𝐴 ∧ (coda‘𝐹) = (doma‘𝐺)))
18	17	anbi2i 624	. . 3 ⊢ ((𝐺 ∈ 𝐴 ∧ 𝐹 ∈ {ℎ ∈ 𝐴 ∣ (coda‘ℎ) = (doma‘𝐺)}) ↔ (𝐺 ∈ 𝐴 ∧ (𝐹 ∈ 𝐴 ∧ (coda‘𝐹) = (doma‘𝐺))))
19		an12 646	. . . 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 1542   ∈ wcel 2114  ∀wral 3052  {crab 3390  Vcvv 3430  {csn 4568  ⟨cop 4574  ⟨cotp 4576  ∪ ciun 4934   class class class wbr 5086   × cxp 5624  dom cdm 5626  ⟶wf 6490  ‘cfv 6494  (class class class)co 7362  2^nd c2nd 7936  compcco 17227  dom_acdoma 17982  cod_accoda 17983  Arrowcarw 17984  comp_accoa 18016

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5213  ax-sep 5232  ax-nul 5242  ax-pow 5304  ax-pr 5372  ax-un 7684

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-ot 4577  df-uni 4852  df-iun 4936  df-br 5087  df-opab 5149  df-mpt 5168  df-id 5521  df-xp 5632  df-rel 5633  df-cnv 5634  df-co 5635  df-dm 5636  df-rn 5637  df-res 5638  df-ima 5639  df-iota 6450  df-fun 6496  df-fn 6497  df-f 6498  df-f1 6499  df-fo 6500  df-f1o 6501  df-fv 6502  df-ov 7365  df-oprab 7366  df-mpo 7367  df-1st 7937  df-2nd 7938  df-arw 17989  df-coa 18018

This theorem is referenced by:  homdmcoa  18029  coapm  18033