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Theorem coapm 18123

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 2763	. . . . . 6 ⊢ (comp‘𝐶) = (comp‘𝐶)
4	1, 2, 3	coafval 18116	. . . . 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 7534	. . . 4 ⊢ Fun ·
6		funfn 6566	. . . 4 ⊢ (Fun · ↔ · Fn dom · )
7	5, 6	mpbi 233	. . 3 ⊢ · Fn dom ·
8	1, 2	dmcoass 18118	. . . . . . . . 9 ⊢ dom · ⊆ (𝐴 × 𝐴)
9	8	sseli 3933	. . . . . . . 8 ⊢ (𝑧 ∈ dom · → 𝑧 ∈ (𝐴 × 𝐴))
10		1st2nd2 8021	. . . . . . . 8 ⊢ (𝑧 ∈ (𝐴 × 𝐴) → 𝑧 = ⟨(1^st ‘𝑧), (2^nd ‘𝑧)⟩)
11	9, 10	syl 18	. . . . . . 7 ⊢ (𝑧 ∈ dom · → 𝑧 = ⟨(1^st ‘𝑧), (2^nd ‘𝑧)⟩)
12	11	fveq2d 6885	. . . . . 6 ⊢ (𝑧 ∈ dom · → ( · ‘𝑧) = ( · ‘⟨(1^st ‘𝑧), (2^nd ‘𝑧)⟩))
13		df-ov 7413	. . . . . 6 ⊢ ((1^st ‘𝑧) · (2^nd ‘𝑧)) = ( · ‘⟨(1^st ‘𝑧), (2^nd ‘𝑧)⟩)
14	12, 13	eqtr4di 2816	. . . . 5 ⊢ (𝑧 ∈ dom · → ( · ‘𝑧) = ((1^st ‘𝑧) · (2^nd ‘𝑧)))
15		eqid 2763	. . . . . . 7 ⊢ (Hom_a‘𝐶) = (Hom_a‘𝐶)
16	2, 15	homarw 18098	. . . . . 6 ⊢ ((dom_a‘(2^nd ‘𝑧))(Hom_a‘𝐶)(cod_a‘(1^st ‘𝑧))) ⊆ 𝐴
17		id 23	. . . . . . . . . . . . 13 ⊢ (𝑧 ∈ dom · → 𝑧 ∈ dom · )
18	11, 17	eqeltrrd 2864	. . . . . . . . . . . 12 ⊢ (𝑧 ∈ dom · → ⟨(1^st ‘𝑧), (2^nd ‘𝑧)⟩ ∈ dom · )
19		df-br 5110	. . . . . . . . . . . 12 ⊢ ((1^st ‘𝑧)dom · (2^nd ‘𝑧) ↔ ⟨(1^st ‘𝑧), (2^nd ‘𝑧)⟩ ∈ dom · )
20	18, 19	sylibr 237	. . . . . . . . . . 11 ⊢ (𝑧 ∈ dom · → (1^st ‘𝑧)dom · (2^nd ‘𝑧))
21	1, 2	eldmcoa 18117	. . . . . . . . . . 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 221	. . . . . . . . . 10 ⊢ (𝑧 ∈ dom · → ((2^nd ‘𝑧) ∈ 𝐴 ∧ (1^st ‘𝑧) ∈ 𝐴 ∧ (cod_a‘(2^nd ‘𝑧)) = (dom_a‘(1^st ‘𝑧))))
23	22	simp1d 1160	. . . . . . . . 9 ⊢ (𝑧 ∈ dom · → (2^nd ‘𝑧) ∈ 𝐴)
24	2, 15	arwhoma 18097	. . . . . . . . 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 18	. . . . . . . 8 ⊢ (𝑧 ∈ dom · → (2^nd ‘𝑧) ∈ ((dom_a‘(2^nd ‘𝑧))(Hom_a‘𝐶)(cod_a‘(2^nd ‘𝑧))))
26	22	simp3d 1162	. . . . . . . . 9 ⊢ (𝑧 ∈ dom · → (cod_a‘(2^nd ‘𝑧)) = (dom_a‘(1^st ‘𝑧)))
27	26	oveq2d 7426	. . . . . . . 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 2865	. . . . . . 7 ⊢ (𝑧 ∈ dom · → (2^nd ‘𝑧) ∈ ((dom_a‘(2^nd ‘𝑧))(Hom_a‘𝐶)(dom_a‘(1^st ‘𝑧))))
29	22	simp2d 1161	. . . . . . . 8 ⊢ (𝑧 ∈ dom · → (1^st ‘𝑧) ∈ 𝐴)
30	2, 15	arwhoma 18097	. . . . . . . 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 18	. . . . . . 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 18122	. . . . . 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 3935	. . . . 5 ⊢ (𝑧 ∈ dom · → ((1^st ‘𝑧) · (2^nd ‘𝑧)) ∈ 𝐴)
34	14, 33	eqeltrd 2863	. . . 4 ⊢ (𝑧 ∈ dom · → ( · ‘𝑧) ∈ 𝐴)
35	34	rgen 3081	. . 3 ⊢ ∀𝑧 ∈ dom · ( · ‘𝑧) ∈ 𝐴
36		ffnfv 7114	. . 3 ⊢ ( · :dom · ⟶𝐴 ↔ ( · Fn dom · ∧ ∀𝑧 ∈ dom · ( · ‘𝑧) ∈ 𝐴))
37	7, 35, 36	mpbir2an 723	. 2 ⊢ · :dom · ⟶𝐴
38	2	fvexi 6895	. . 3 ⊢ 𝐴 ∈ V
39	38, 38	xpex 7748	. . 3 ⊢ (𝐴 × 𝐴) ∈ V
40	38, 39	elpm2 8868	. 2 ⊢ ( · ∈ (𝐴 ↑_pm (𝐴 × 𝐴)) ↔ ( · :dom · ⟶𝐴 ∧ dom · ⊆ (𝐴 × 𝐴)))
41	37, 8, 40	mpbir2an 723	1 ⊢ · ∈ (𝐴 ↑_pm (𝐴 × 𝐴))

Colors of variables: wff setvar class

Syntax hints: ∧ w3a 1103 = wceq 1570 ∈ wcel 2143 ∀wral 3079 {crab 3416 ⊆ wss 3905 ⟨cop 4595 ⟨cotp 4597 class class class wbr 5109 × cxp 5659 dom cdm 5661 Fun wfun 6530 Fn wfn 6531 ⟶wf 6532 ‘cfv 6536 (class class class)co 7410 1^st c1st 7980 2^nd c2nd 7981 ↑_pm cpm 8821 compcco 17317 dom_acdoma 18072 cod_accoda 18073 Arrowcarw 18074 Hom_achoma 18075 comp_accoa 18106

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1825 ax-4 1839 ax-5 1940 ax-6 1997 ax-7 2038 ax-8 2145 ax-9 2153 ax-10 2176 ax-11 2192 ax-12 2213 ax-ext 2735 ax-rep 5238 ax-sep 5257 ax-nul 5269 ax-pow 5336 ax-pr 5404 ax-un 7732

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1810 df-nf 1814 df-sb 2097 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2838 df-nfc 2912 df-ne 2959 df-ral 3080 df-rex 3090 df-reu 3370 df-rab 3417 df-v 3457 df-sbc 3745 df-csb 3854 df-dif 3908 df-un 3910 df-in 3912 df-ss 3922 df-nul 4287 df-if 4488 df-pw 4564 df-sn 4590 df-pr 4592 df-op 4596 df-ot 4598 df-uni 4873 df-iun 4958 df-br 5110 df-opab 5174 df-mpt 5193 df-id 5556 df-xp 5667 df-rel 5668 df-cnv 5669 df-co 5670 df-dm 5671 df-rn 5672 df-res 5673 df-ima 5674 df-iota 6492 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-ov 7413 df-oprab 7414 df-mpo 7415 df-1st 7982 df-2nd 7983 df-pm 8823 df-cat 17719 df-doma 18076 df-coda 18077 df-homa 18078 df-arw 18079 df-coa 18108

This theorem is referenced by: (None)

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