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Theorem coapm 18138

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 2740	. . . . . 6 ⊢ (comp‘𝐶) = (comp‘𝐶)
4	1, 2, 3	coafval 18131	. . . . 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 7574	. . . 4 ⊢ Fun ·
6		funfn 6608	. . . 4 ⊢ (Fun · ↔ · Fn dom · )
7	5, 6	mpbi 230	. . 3 ⊢ · Fn dom ·
8	1, 2	dmcoass 18133	. . . . . . . . 9 ⊢ dom · ⊆ (𝐴 × 𝐴)
9	8	sseli 4004	. . . . . . . 8 ⊢ (𝑧 ∈ dom · → 𝑧 ∈ (𝐴 × 𝐴))
10		1st2nd2 8069	. . . . . . . 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 6924	. . . . . 6 ⊢ (𝑧 ∈ dom · → ( · ‘𝑧) = ( · ‘⟨(1^st ‘𝑧), (2^nd ‘𝑧)⟩))
13		df-ov 7451	. . . . . 6 ⊢ ((1^st ‘𝑧) · (2^nd ‘𝑧)) = ( · ‘⟨(1^st ‘𝑧), (2^nd ‘𝑧)⟩)
14	12, 13	eqtr4di 2798	. . . . 5 ⊢ (𝑧 ∈ dom · → ( · ‘𝑧) = ((1^st ‘𝑧) · (2^nd ‘𝑧)))
15		eqid 2740	. . . . . . 7 ⊢ (Hom_a‘𝐶) = (Hom_a‘𝐶)
16	2, 15	homarw 18113	. . . . . 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 2845	. . . . . . . . . . . 12 ⊢ (𝑧 ∈ dom · → ⟨(1^st ‘𝑧), (2^nd ‘𝑧)⟩ ∈ dom · )
19		df-br 5167	. . . . . . . . . . . 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 18132	. . . . . . . . . . 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 1142	. . . . . . . . 9 ⊢ (𝑧 ∈ dom · → (2^nd ‘𝑧) ∈ 𝐴)
24	2, 15	arwhoma 18112	. . . . . . . . 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 1144	. . . . . . . . 9 ⊢ (𝑧 ∈ dom · → (cod_a‘(2^nd ‘𝑧)) = (dom_a‘(1^st ‘𝑧)))
27	26	oveq2d 7464	. . . . . . . 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 2846	. . . . . . 7 ⊢ (𝑧 ∈ dom · → (2^nd ‘𝑧) ∈ ((dom_a‘(2^nd ‘𝑧))(Hom_a‘𝐶)(dom_a‘(1^st ‘𝑧))))
29	22	simp2d 1143	. . . . . . . 8 ⊢ (𝑧 ∈ dom · → (1^st ‘𝑧) ∈ 𝐴)
30	2, 15	arwhoma 18112	. . . . . . . 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 18137	. . . . . 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 4006	. . . . 5 ⊢ (𝑧 ∈ dom · → ((1^st ‘𝑧) · (2^nd ‘𝑧)) ∈ 𝐴)
34	14, 33	eqeltrd 2844	. . . 4 ⊢ (𝑧 ∈ dom · → ( · ‘𝑧) ∈ 𝐴)
35	34	rgen 3069	. . 3 ⊢ ∀𝑧 ∈ dom · ( · ‘𝑧) ∈ 𝐴
36		ffnfv 7153	. . 3 ⊢ ( · :dom · ⟶𝐴 ↔ ( · Fn dom · ∧ ∀𝑧 ∈ dom · ( · ‘𝑧) ∈ 𝐴))
37	7, 35, 36	mpbir2an 710	. 2 ⊢ · :dom · ⟶𝐴
38	2	fvexi 6934	. . 3 ⊢ 𝐴 ∈ V
39	38, 38	xpex 7788	. . 3 ⊢ (𝐴 × 𝐴) ∈ V
40	38, 39	elpm2 8932	. 2 ⊢ ( · ∈ (𝐴 ↑_pm (𝐴 × 𝐴)) ↔ ( · :dom · ⟶𝐴 ∧ dom · ⊆ (𝐴 × 𝐴)))
41	37, 8, 40	mpbir2an 710	1 ⊢ · ∈ (𝐴 ↑_pm (𝐴 × 𝐴))

Colors of variables: wff setvar class

Syntax hints: ∧ w3a 1087 = wceq 1537 ∈ wcel 2108 ∀wral 3067 {crab 3443 ⊆ wss 3976 ⟨cop 4654 ⟨cotp 4656 class class class wbr 5166 × cxp 5698 dom cdm 5700 Fun wfun 6567 Fn wfn 6568 ⟶wf 6569 ‘cfv 6573 (class class class)co 7448 1^st c1st 8028 2^nd c2nd 8029 ↑_pm cpm 8885 compcco 17323 dom_acdoma 18087 cod_accoda 18088 Arrowcarw 18089 Hom_achoma 18090 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-pm 8887 df-cat 17726 df-doma 18091 df-coda 18092 df-homa 18093 df-arw 18094 df-coa 18123

This theorem is referenced by: (None)

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