eldmcoa - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > eldmcoa		Structured version Visualization version GIF version

Theorem eldmcoa 18015

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	⊢ · = (comp_a‘𝐶)
coafval.a	⊢ 𝐴 = (Arrow‘𝐶)

**Assertion**
Ref	Expression
eldmcoa	⊢ (𝐺dom · 𝐹 ↔ (𝐹 ∈ 𝐴 ∧ 𝐺 ∈ 𝐴 ∧ (cod_a‘𝐹) = (dom_a‘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 5150	. 2 ⊢ (𝐺dom · 𝐹 ↔ ⟨𝐺, 𝐹⟩ ∈ dom · )
2		otex 5466	. . . . . 6 ⊢ ⟨(dom_a‘𝑓), (cod_a‘𝑔), ((2^nd ‘𝑔)(⟨(dom_a‘𝑓), (dom_a‘𝑔)⟩(comp‘𝐶)(cod_a‘𝑔))(2^nd ‘𝑓))⟩ ∈ V
3	2	rgen2w 3067	. . . . 5 ⊢ ∀𝑔 ∈ 𝐴 ∀𝑓 ∈ {ℎ ∈ 𝐴 ∣ (cod_a‘ℎ) = (dom_a‘𝑔)}⟨(dom_a‘𝑓), (cod_a‘𝑔), ((2^nd ‘𝑔)(⟨(dom_a‘𝑓), (dom_a‘𝑔)⟩(comp‘𝐶)(cod_a‘𝑔))(2^nd ‘𝑓))⟩ ∈ V
4		coafval.o	. . . . . . 7 ⊢ · = (comp_a‘𝐶)
5		coafval.a	. . . . . . 7 ⊢ 𝐴 = (Arrow‘𝐶)
6		eqid 2733	. . . . . . 7 ⊢ (comp‘𝐶) = (comp‘𝐶)
7	4, 5, 6	coafval 18014	. . . . . 6 ⊢ · = (𝑔 ∈ 𝐴, 𝑓 ∈ {ℎ ∈ 𝐴 ∣ (cod_a‘ℎ) = (dom_a‘𝑔)} ↦ ⟨(dom_a‘𝑓), (cod_a‘𝑔), ((2^nd ‘𝑔)(⟨(dom_a‘𝑓), (dom_a‘𝑔)⟩(comp‘𝐶)(cod_a‘𝑔))(2^nd ‘𝑓))⟩)
8	7	fmpox 8053	. . . . 5 ⊢ (∀𝑔 ∈ 𝐴 ∀𝑓 ∈ {ℎ ∈ 𝐴 ∣ (cod_a‘ℎ) = (dom_a‘𝑔)}⟨(dom_a‘𝑓), (cod_a‘𝑔), ((2^nd ‘𝑔)(⟨(dom_a‘𝑓), (dom_a‘𝑔)⟩(comp‘𝐶)(cod_a‘𝑔))(2^nd ‘𝑓))⟩ ∈ V ↔ · :∪ 𝑔 ∈ 𝐴 ({𝑔} × {ℎ ∈ 𝐴 ∣ (cod_a‘ℎ) = (dom_a‘𝑔)})⟶V)
9	3, 8	mpbi 229	. . . 4 ⊢ · :∪ 𝑔 ∈ 𝐴 ({𝑔} × {ℎ ∈ 𝐴 ∣ (cod_a‘ℎ) = (dom_a‘𝑔)})⟶V
10	9	fdmi 6730	. . 3 ⊢ dom · = ∪ 𝑔 ∈ 𝐴 ({𝑔} × {ℎ ∈ 𝐴 ∣ (cod_a‘ℎ) = (dom_a‘𝑔)})
11	10	eleq2i 2826	. 2 ⊢ (⟨𝐺, 𝐹⟩ ∈ dom · ↔ ⟨𝐺, 𝐹⟩ ∈ ∪ 𝑔 ∈ 𝐴 ({𝑔} × {ℎ ∈ 𝐴 ∣ (cod_a‘ℎ) = (dom_a‘𝑔)}))
12		fveq2 6892	. . . . . 6 ⊢ (𝑔 = 𝐺 → (dom_a‘𝑔) = (dom_a‘𝐺))
13	12	eqeq2d 2744	. . . . 5 ⊢ (𝑔 = 𝐺 → ((cod_a‘ℎ) = (dom_a‘𝑔) ↔ (cod_a‘ℎ) = (dom_a‘𝐺)))
14	13	rabbidv 3441	. . . 4 ⊢ (𝑔 = 𝐺 → {ℎ ∈ 𝐴 ∣ (cod_a‘ℎ) = (dom_a‘𝑔)} = {ℎ ∈ 𝐴 ∣ (cod_a‘ℎ) = (dom_a‘𝐺)})
15	14	opeliunxp2 5839	. . 3 ⊢ (⟨𝐺, 𝐹⟩ ∈ ∪ 𝑔 ∈ 𝐴 ({𝑔} × {ℎ ∈ 𝐴 ∣ (cod_a‘ℎ) = (dom_a‘𝑔)}) ↔ (𝐺 ∈ 𝐴 ∧ 𝐹 ∈ {ℎ ∈ 𝐴 ∣ (cod_a‘ℎ) = (dom_a‘𝐺)}))
16		fveqeq2 6901	. . . . 5 ⊢ (ℎ = 𝐹 → ((cod_a‘ℎ) = (dom_a‘𝐺) ↔ (cod_a‘𝐹) = (dom_a‘𝐺)))
17	16	elrab 3684	. . . 4 ⊢ (𝐹 ∈ {ℎ ∈ 𝐴 ∣ (cod_a‘ℎ) = (dom_a‘𝐺)} ↔ (𝐹 ∈ 𝐴 ∧ (cod_a‘𝐹) = (dom_a‘𝐺)))
18	17	anbi2i 624	. . 3 ⊢ ((𝐺 ∈ 𝐴 ∧ 𝐹 ∈ {ℎ ∈ 𝐴 ∣ (cod_a‘ℎ) = (dom_a‘𝐺)}) ↔ (𝐺 ∈ 𝐴 ∧ (𝐹 ∈ 𝐴 ∧ (cod_a‘𝐹) = (dom_a‘𝐺))))
19		an12 644	. . . 4 ⊢ ((𝐺 ∈ 𝐴 ∧ (𝐹 ∈ 𝐴 ∧ (cod_a‘𝐹) = (dom_a‘𝐺))) ↔ (𝐹 ∈ 𝐴 ∧ (𝐺 ∈ 𝐴 ∧ (cod_a‘𝐹) = (dom_a‘𝐺))))
20		3anass 1096	. . . 4 ⊢ ((𝐹 ∈ 𝐴 ∧ 𝐺 ∈ 𝐴 ∧ (cod_a‘𝐹) = (dom_a‘𝐺)) ↔ (𝐹 ∈ 𝐴 ∧ (𝐺 ∈ 𝐴 ∧ (cod_a‘𝐹) = (dom_a‘𝐺))))
21	19, 20	bitr4i 278	. . 3 ⊢ ((𝐺 ∈ 𝐴 ∧ (𝐹 ∈ 𝐴 ∧ (cod_a‘𝐹) = (dom_a‘𝐺))) ↔ (𝐹 ∈ 𝐴 ∧ 𝐺 ∈ 𝐴 ∧ (cod_a‘𝐹) = (dom_a‘𝐺)))
22	15, 18, 21	3bitri 297	. 2 ⊢ (⟨𝐺, 𝐹⟩ ∈ ∪ 𝑔 ∈ 𝐴 ({𝑔} × {ℎ ∈ 𝐴 ∣ (cod_a‘ℎ) = (dom_a‘𝑔)}) ↔ (𝐹 ∈ 𝐴 ∧ 𝐺 ∈ 𝐴 ∧ (cod_a‘𝐹) = (dom_a‘𝐺)))
23	1, 11, 22	3bitri 297	1 ⊢ (𝐺dom · 𝐹 ↔ (𝐹 ∈ 𝐴 ∧ 𝐺 ∈ 𝐴 ∧ (cod_a‘𝐹) = (dom_a‘𝐺)))

Colors of variables: wff setvar class

Syntax hints: ↔ wb 205 ∧ wa 397 ∧ w3a 1088 = wceq 1542 ∈ wcel 2107 ∀wral 3062 {crab 3433 Vcvv 3475 {csn 4629 ⟨cop 4635 ⟨cotp 4637 ∪ ciun 4998 class class class wbr 5149 × cxp 5675 dom cdm 5677 ⟶wf 6540 ‘cfv 6544 (class class class)co 7409 2^nd c2nd 7974 compcco 17209 dom_acdoma 17970 cod_accoda 17971 Arrowcarw 17972 comp_accoa 18004

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7725

This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-ral 3063 df-rex 3072 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-ot 4638 df-uni 4910 df-iun 5000 df-br 5150 df-opab 5212 df-mpt 5233 df-id 5575 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-ov 7412 df-oprab 7413 df-mpo 7414 df-1st 7975 df-2nd 7976 df-arw 17977 df-coa 18006

This theorem is referenced by: homdmcoa 18017 coapm 18021

W3C validator