coapm - Metamath Proof Explorer

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

Theorem coapm 18116

Description: Composition of arrows is a partial binary operation on arrows. (Contributed by Mario Carneiro, 11-Jan-2017.)

**Hypotheses**
Ref	Expression
coapm.o	⊢ · = (comp_a‘𝐶)
coapm.a	⊢ 𝐴 = (Arrow‘𝐶)

**Assertion**
Ref	Expression
coapm	⊢ · ∈ (𝐴 ↑_pm (𝐴 × 𝐴))

Proof of Theorem coapm Dummy variables 𝑓 𝑔 ℎ 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		coapm.o	. . . . . 6 ⊢ · = (comp_a‘𝐶)
2		coapm.a	. . . . . 6 ⊢ 𝐴 = (Arrow‘𝐶)
3		eqid 2737	. . . . . 6 ⊢ (comp‘𝐶) = (comp‘𝐶)
4	1, 2, 3	coafval 18109	. . . . 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))⟩)
5	4	mpofun 7557	. . . 4 ⊢ Fun ·
6		funfn 6596	. . . 4 ⊢ (Fun · ↔ · Fn dom · )
7	5, 6	mpbi 230	. . 3 ⊢ · Fn dom ·
8	1, 2	dmcoass 18111	. . . . . . . . 9 ⊢ dom · ⊆ (𝐴 × 𝐴)
9	8	sseli 3979	. . . . . . . 8 ⊢ (𝑧 ∈ dom · → 𝑧 ∈ (𝐴 × 𝐴))
10		1st2nd2 8053	. . . . . . . 8 ⊢ (𝑧 ∈ (𝐴 × 𝐴) → 𝑧 = ⟨(1^st ‘𝑧), (2^nd ‘𝑧)⟩)
11	9, 10	syl 17	. . . . . . 7 ⊢ (𝑧 ∈ dom · → 𝑧 = ⟨(1^st ‘𝑧), (2^nd ‘𝑧)⟩)
12	11	fveq2d 6910	. . . . . 6 ⊢ (𝑧 ∈ dom · → ( · ‘𝑧) = ( · ‘⟨(1^st ‘𝑧), (2^nd ‘𝑧)⟩))
13		df-ov 7434	. . . . . 6 ⊢ ((1^st ‘𝑧) · (2^nd ‘𝑧)) = ( · ‘⟨(1^st ‘𝑧), (2^nd ‘𝑧)⟩)
14	12, 13	eqtr4di 2795	. . . . 5 ⊢ (𝑧 ∈ dom · → ( · ‘𝑧) = ((1^st ‘𝑧) · (2^nd ‘𝑧)))
15		eqid 2737	. . . . . . 7 ⊢ (Hom_a‘𝐶) = (Hom_a‘𝐶)
16	2, 15	homarw 18091	. . . . . 6 ⊢ ((dom_a‘(2^nd ‘𝑧))(Hom_a‘𝐶)(cod_a‘(1^st ‘𝑧))) ⊆ 𝐴
17		id 22	. . . . . . . . . . . . 13 ⊢ (𝑧 ∈ dom · → 𝑧 ∈ dom · )
18	11, 17	eqeltrrd 2842	. . . . . . . . . . . 12 ⊢ (𝑧 ∈ dom · → ⟨(1^st ‘𝑧), (2^nd ‘𝑧)⟩ ∈ dom · )
19		df-br 5144	. . . . . . . . . . . 12 ⊢ ((1^st ‘𝑧)dom · (2^nd ‘𝑧) ↔ ⟨(1^st ‘𝑧), (2^nd ‘𝑧)⟩ ∈ dom · )
20	18, 19	sylibr 234	. . . . . . . . . . 11 ⊢ (𝑧 ∈ dom · → (1^st ‘𝑧)dom · (2^nd ‘𝑧))
21	1, 2	eldmcoa 18110	. . . . . . . . . . 11 ⊢ ((1^st ‘𝑧)dom · (2^nd ‘𝑧) ↔ ((2^nd ‘𝑧) ∈ 𝐴 ∧ (1^st ‘𝑧) ∈ 𝐴 ∧ (cod_a‘(2^nd ‘𝑧)) = (dom_a‘(1^st ‘𝑧))))
22	20, 21	sylib 218	. . . . . . . . . 10 ⊢ (𝑧 ∈ dom · → ((2^nd ‘𝑧) ∈ 𝐴 ∧ (1^st ‘𝑧) ∈ 𝐴 ∧ (cod_a‘(2^nd ‘𝑧)) = (dom_a‘(1^st ‘𝑧))))
23	22	simp1d 1143	. . . . . . . . 9 ⊢ (𝑧 ∈ dom · → (2^nd ‘𝑧) ∈ 𝐴)
24	2, 15	arwhoma 18090	. . . . . . . . 9 ⊢ ((2^nd ‘𝑧) ∈ 𝐴 → (2^nd ‘𝑧) ∈ ((dom_a‘(2^nd ‘𝑧))(Hom_a‘𝐶)(cod_a‘(2^nd ‘𝑧))))
25	23, 24	syl 17	. . . . . . . 8 ⊢ (𝑧 ∈ dom · → (2^nd ‘𝑧) ∈ ((dom_a‘(2^nd ‘𝑧))(Hom_a‘𝐶)(cod_a‘(2^nd ‘𝑧))))
26	22	simp3d 1145	. . . . . . . . 9 ⊢ (𝑧 ∈ dom · → (cod_a‘(2^nd ‘𝑧)) = (dom_a‘(1^st ‘𝑧)))
27	26	oveq2d 7447	. . . . . . . 8 ⊢ (𝑧 ∈ dom · → ((dom_a‘(2^nd ‘𝑧))(Hom_a‘𝐶)(cod_a‘(2^nd ‘𝑧))) = ((dom_a‘(2^nd ‘𝑧))(Hom_a‘𝐶)(dom_a‘(1^st ‘𝑧))))
28	25, 27	eleqtrd 2843	. . . . . . 7 ⊢ (𝑧 ∈ dom · → (2^nd ‘𝑧) ∈ ((dom_a‘(2^nd ‘𝑧))(Hom_a‘𝐶)(dom_a‘(1^st ‘𝑧))))
29	22	simp2d 1144	. . . . . . . 8 ⊢ (𝑧 ∈ dom · → (1^st ‘𝑧) ∈ 𝐴)
30	2, 15	arwhoma 18090	. . . . . . . 8 ⊢ ((1^st ‘𝑧) ∈ 𝐴 → (1^st ‘𝑧) ∈ ((dom_a‘(1^st ‘𝑧))(Hom_a‘𝐶)(cod_a‘(1^st ‘𝑧))))
31	29, 30	syl 17	. . . . . . 7 ⊢ (𝑧 ∈ dom · → (1^st ‘𝑧) ∈ ((dom_a‘(1^st ‘𝑧))(Hom_a‘𝐶)(cod_a‘(1^st ‘𝑧))))
32	1, 15, 28, 31	coahom 18115	. . . . . 6 ⊢ (𝑧 ∈ dom · → ((1^st ‘𝑧) · (2^nd ‘𝑧)) ∈ ((dom_a‘(2^nd ‘𝑧))(Hom_a‘𝐶)(cod_a‘(1^st ‘𝑧))))
33	16, 32	sselid 3981	. . . . 5 ⊢ (𝑧 ∈ dom · → ((1^st ‘𝑧) · (2^nd ‘𝑧)) ∈ 𝐴)
34	14, 33	eqeltrd 2841	. . . 4 ⊢ (𝑧 ∈ dom · → ( · ‘𝑧) ∈ 𝐴)
35	34	rgen 3063	. . 3 ⊢ ∀𝑧 ∈ dom · ( · ‘𝑧) ∈ 𝐴
36		ffnfv 7139	. . 3 ⊢ ( · :dom · ⟶𝐴 ↔ ( · Fn dom · ∧ ∀𝑧 ∈ dom · ( · ‘𝑧) ∈ 𝐴))
37	7, 35, 36	mpbir2an 711	. 2 ⊢ · :dom · ⟶𝐴
38	2	fvexi 6920	. . 3 ⊢ 𝐴 ∈ V
39	38, 38	xpex 7773	. . 3 ⊢ (𝐴 × 𝐴) ∈ V
40	38, 39	elpm2 8914	. 2 ⊢ ( · ∈ (𝐴 ↑_pm (𝐴 × 𝐴)) ↔ ( · :dom · ⟶𝐴 ∧ dom · ⊆ (𝐴 × 𝐴)))
41	37, 8, 40	mpbir2an 711	1 ⊢ · ∈ (𝐴 ↑_pm (𝐴 × 𝐴))

Colors of variables: wff setvar class

Syntax hints: ∧ w3a 1087 = wceq 1540 ∈ wcel 2108 ∀wral 3061 {crab 3436 ⊆ wss 3951 ⟨cop 4632 ⟨cotp 4634 class class class wbr 5143 × cxp 5683 dom cdm 5685 Fun wfun 6555 Fn wfn 6556 ⟶wf 6557 ‘cfv 6561 (class class class)co 7431 1^st c1st 8012 2^nd c2nd 8013 ↑_pm cpm 8867 compcco 17309 dom_acdoma 18065 cod_accoda 18066 Arrowcarw 18067 Hom_achoma 18068 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-pm 8869 df-cat 17711 df-doma 18069 df-coda 18070 df-homa 18071 df-arw 18072 df-coa 18101

This theorem is referenced by: (None)

W3C validator