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Theorem coapm 18246

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 2761	. . . . . 6 ⊢ (comp‘𝐶) = (comp‘𝐶)
4	1, 2, 3	coafval 18239	. . . . 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 7544	. . . 4 ⊢ Fun ·
6		funfn 6570	. . . 4 ⊢ (Fun · ↔ · Fn dom · )
7	5, 6	mpbi 233	. . 3 ⊢ · Fn dom ·
8	1, 2	dmcoass 18241	. . . . . . . . 9 ⊢ dom · ⊆ (𝐴 × 𝐴)
9	8	sseli 3927	. . . . . . . 8 ⊢ (𝑧 ∈ dom · → 𝑧 ∈ (𝐴 × 𝐴))
10		1st2nd2 8040	. . . . . . . 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 6889	. . . . . 6 ⊢ (𝑧 ∈ dom · → ( · ‘𝑧) = ( · ‘⟨(1^st ‘𝑧), (2^nd ‘𝑧)⟩))
13		df-ov 7423	. . . . . 6 ⊢ ((1^st ‘𝑧) · (2^nd ‘𝑧)) = ( · ‘⟨(1^st ‘𝑧), (2^nd ‘𝑧)⟩)
14	12, 13	eqtr4di 2814	. . . . 5 ⊢ (𝑧 ∈ dom · → ( · ‘𝑧) = ((1^st ‘𝑧) · (2^nd ‘𝑧)))
15		eqid 2761	. . . . . . 7 ⊢ (Hom_a‘𝐶) = (Hom_a‘𝐶)
16	2, 15	homarw 18221	. . . . . 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 2862	. . . . . . . . . . . 12 ⊢ (𝑧 ∈ dom · → ⟨(1^st ‘𝑧), (2^nd ‘𝑧)⟩ ∈ dom · )
19		df-br 5104	. . . . . . . . . . . 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 18240	. . . . . . . . . . 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 18220	. . . . . . . . 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 7436	. . . . . . . 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 2863	. . . . . . 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 18220	. . . . . . . 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 18245	. . . . . 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 3929	. . . . 5 ⊢ (𝑧 ∈ dom · → ((1^st ‘𝑧) · (2^nd ‘𝑧)) ∈ 𝐴)
34	14, 33	eqeltrd 2861	. . . 4 ⊢ (𝑧 ∈ dom · → ( · ‘𝑧) ∈ 𝐴)
35	34	rgen 3079	. . 3 ⊢ ∀𝑧 ∈ dom · ( · ‘𝑧) ∈ 𝐴
36		ffnfv 7119	. . 3 ⊢ ( · :dom · ⟶𝐴 ↔ ( · Fn dom · ∧ ∀𝑧 ∈ dom · ( · ‘𝑧) ∈ 𝐴))
37	7, 35, 36	mpbir2an 724	. 2 ⊢ · :dom · ⟶𝐴
38	2	fvexi 6899	. . 3 ⊢ 𝐴 ∈ V
39	38, 38	xpex 7767	. . 3 ⊢ (𝐴 × 𝐴) ∈ V
40	38, 39	elpm2 8902	. 2 ⊢ ( · ∈ (𝐴 ↑_pm (𝐴 × 𝐴)) ↔ ( · :dom · ⟶𝐴 ∧ dom · ⊆ (𝐴 × 𝐴)))
41	37, 8, 40	mpbir2an 724	1 ⊢ · ∈ (𝐴 ↑_pm (𝐴 × 𝐴))

Colors of variables: wff setvar class

This proof depends on syntax axioms: ∧ w3a 1103 = wceq 1570 ∈ wcel 2145 ∀wral 3077 {crab 3413 ⊆ wss 3899 ⟨cop 4590 ⟨cotp 4592 class class class wbr 5103 × cxp 5649 dom cdm 5651 Fun wfun 6532 Fn wfn 6533 ⟶wf 6534 ‘cfv 6538 (class class class)co 7420 1^st c1st 7999 2^nd c2nd 8000 ↑_pm cpm 8848 compcco 17440 dom_acdoma 18195 cod_accoda 18196 Arrowcarw 18197 Hom_achoma 18198 comp_accoa 18229

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2213 ax-ext 2733 ax-rep 5232 ax-sep 5249 ax-nul 5260 ax-pow 5327 ax-pr 5391 ax-un 7751

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2565 df-eu 2595 df-clab 2740 df-cleq 2753 df-clel 2836 df-nfc 2910 df-ne 2957 df-ral 3078 df-rex 3088 df-reu 3367 df-rab 3414 df-v 3453 df-sbc 3740 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-nul 4280 df-if 4483 df-pw 4559 df-sn 4585 df-pr 4587 df-op 4591 df-ot 4593 df-uni 4868 df-iun 4953 df-br 5104 df-opab 5168 df-mpt 5187 df-id 5546 df-xp 5657 df-rel 5658 df-cnv 5659 df-co 5660 df-dm 5661 df-rn 5662 df-res 5663 df-ima 5664 df-iota 6494 df-fun 6540 df-fn 6541 df-f 6542 df-f1 6543 df-fo 6544 df-f1o 6545 df-fv 6546 df-ov 7423 df-oprab 7424 df-mpo 7425 df-1st 8001 df-2nd 8002 df-pm 8850 df-cat 17842 df-doma 18199 df-coda 18200 df-homa 18201 df-arw 18202 df-coa 18231

This theorem is used by: (None)

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