Theorem eldmcoa 18132

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 5167	. 2 ⊢ (𝐺dom · 𝐹 ↔ ⟨𝐺, 𝐹⟩ ∈ dom · )
2		otex 5485	. . . . . 6 ⊢ ⟨(doma‘𝑓), (coda‘𝑔), ((2nd ‘𝑔)(⟨(doma‘𝑓), (doma‘𝑔)⟩(comp‘𝐶)(coda‘𝑔))(2nd ‘𝑓))⟩ ∈ V
3	2	rgen2w 3072	. . . . 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 2740	. . . . . . 7 ⊢ (comp‘𝐶) = (comp‘𝐶)
7	4, 5, 6	coafval 18131	. . . . . 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 8108	. . . . 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 6758	. . 3 ⊢ dom · = ∪ 𝑔 ∈ 𝐴 ({𝑔} × {ℎ ∈ 𝐴 ∣ (coda‘ℎ) = (doma‘𝑔)})
11	10	eleq2i 2836	. 2 ⊢ (⟨𝐺, 𝐹⟩ ∈ dom · ↔ ⟨𝐺, 𝐹⟩ ∈ ∪ 𝑔 ∈ 𝐴 ({𝑔} × {ℎ ∈ 𝐴 ∣ (coda‘ℎ) = (doma‘𝑔)}))
12		fveq2 6920	. . . . . 6 ⊢ (𝑔 = 𝐺 → (doma‘𝑔) = (doma‘𝐺))
13	12	eqeq2d 2751	. . . . 5 ⊢ (𝑔 = 𝐺 → ((coda‘ℎ) = (doma‘𝑔) ↔ (coda‘ℎ) = (doma‘𝐺)))
14	13	rabbidv 3451	. . . 4 ⊢ (𝑔 = 𝐺 → {ℎ ∈ 𝐴 ∣ (coda‘ℎ) = (doma‘𝑔)} = {ℎ ∈ 𝐴 ∣ (coda‘ℎ) = (doma‘𝐺)})
15	14	opeliunxp2 5863	. . 3 ⊢ (⟨𝐺, 𝐹⟩ ∈ ∪ 𝑔 ∈ 𝐴 ({𝑔} × {ℎ ∈ 𝐴 ∣ (coda‘ℎ) = (doma‘𝑔)}) ↔ (𝐺 ∈ 𝐴 ∧ 𝐹 ∈ {ℎ ∈ 𝐴 ∣ (coda‘ℎ) = (doma‘𝐺)}))
16		fveqeq2 6929	. . . . 5 ⊢ (ℎ = 𝐹 → ((coda‘ℎ) = (doma‘𝐺) ↔ (coda‘𝐹) = (doma‘𝐺)))
17	16	elrab 3708	. . . 4 ⊢ (𝐹 ∈ {ℎ ∈ 𝐴 ∣ (coda‘ℎ) = (doma‘𝐺)} ↔ (𝐹 ∈ 𝐴 ∧ (coda‘𝐹) = (doma‘𝐺)))
18	17	anbi2i 622	. . 3 ⊢ ((𝐺 ∈ 𝐴 ∧ 𝐹 ∈ {ℎ ∈ 𝐴 ∣ (coda‘ℎ) = (doma‘𝐺)}) ↔ (𝐺 ∈ 𝐴 ∧ (𝐹 ∈ 𝐴 ∧ (coda‘𝐹) = (doma‘𝐺))))
19		an12 644	. . . 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 1537   ∈ wcel 2108  ∀wral 3067  {crab 3443  Vcvv 3488  {csn 4648  ⟨cop 4654  ⟨cotp 4656  ∪ ciun 5015   class class class wbr 5166   × cxp 5698  dom cdm 5700  ⟶wf 6569  ‘cfv 6573  (class class class)co 7448  2^nd c2nd 8029  compcco 17323  dom_acdoma 18087  cod_accoda 18088  Arrowcarw 18089  comp_accoa 18121

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-rep 5303  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-ot 4657  df-uni 4932  df-iun 5017  df-br 5167  df-opab 5229  df-mpt 5250  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-ov 7451  df-oprab 7452  df-mpo 7453  df-1st 8030  df-2nd 8031  df-arw 18094  df-coa 18123

This theorem is referenced by:  homdmcoa  18134  coapm  18138