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Theorem coapm 17436

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 2738	. . . . . 6 ⊢ (comp‘𝐶) = (comp‘𝐶)
4	1, 2, 3	coafval 17429	. . . . 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 7284	. . . 4 ⊢ Fun ·
6		funfn 6363	. . . 4 ⊢ (Fun · ↔ · Fn dom · )
7	5, 6	mpbi 233	. . 3 ⊢ · Fn dom ·
8	1, 2	dmcoass 17431	. . . . . . . . 9 ⊢ dom · ⊆ (𝐴 × 𝐴)
9	8	sseli 3871	. . . . . . . 8 ⊢ (𝑧 ∈ dom · → 𝑧 ∈ (𝐴 × 𝐴))
10		1st2nd2 7746	. . . . . . . 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 6672	. . . . . 6 ⊢ (𝑧 ∈ dom · → ( · ‘𝑧) = ( · ‘⟨(1^st ‘𝑧), (2^nd ‘𝑧)⟩))
13		df-ov 7167	. . . . . 6 ⊢ ((1^st ‘𝑧) · (2^nd ‘𝑧)) = ( · ‘⟨(1^st ‘𝑧), (2^nd ‘𝑧)⟩)
14	12, 13	eqtr4di 2791	. . . . 5 ⊢ (𝑧 ∈ dom · → ( · ‘𝑧) = ((1^st ‘𝑧) · (2^nd ‘𝑧)))
15		eqid 2738	. . . . . . 7 ⊢ (Hom_a‘𝐶) = (Hom_a‘𝐶)
16	2, 15	homarw 17411	. . . . . 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 2834	. . . . . . . . . . . 12 ⊢ (𝑧 ∈ dom · → ⟨(1^st ‘𝑧), (2^nd ‘𝑧)⟩ ∈ dom · )
19		df-br 5028	. . . . . . . . . . . 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 17430	. . . . . . . . . . 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 1143	. . . . . . . . 9 ⊢ (𝑧 ∈ dom · → (2^nd ‘𝑧) ∈ 𝐴)
24	2, 15	arwhoma 17410	. . . . . . . . 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 7180	. . . . . . . 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 2835	. . . . . . 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 17410	. . . . . . . 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 17435	. . . . . 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	sseldi 3873	. . . . 5 ⊢ (𝑧 ∈ dom · → ((1^st ‘𝑧) · (2^nd ‘𝑧)) ∈ 𝐴)
34	14, 33	eqeltrd 2833	. . . 4 ⊢ (𝑧 ∈ dom · → ( · ‘𝑧) ∈ 𝐴)
35	34	rgen 3063	. . 3 ⊢ ∀𝑧 ∈ dom · ( · ‘𝑧) ∈ 𝐴
36		ffnfv 6886	. . 3 ⊢ ( · :dom · ⟶𝐴 ↔ ( · Fn dom · ∧ ∀𝑧 ∈ dom · ( · ‘𝑧) ∈ 𝐴))
37	7, 35, 36	mpbir2an 711	. 2 ⊢ · :dom · ⟶𝐴
38	2	fvexi 6682	. . 3 ⊢ 𝐴 ∈ V
39	38, 38	xpex 7488	. . 3 ⊢ (𝐴 × 𝐴) ∈ V
40	38, 39	elpm2 8477	. 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 1088 = wceq 1542 ∈ wcel 2113 ∀wral 3053 {crab 3057 ⊆ wss 3841 ⟨cop 4519 ⟨cotp 4521 class class class wbr 5027 × cxp 5517 dom cdm 5519 Fun wfun 6327 Fn wfn 6328 ⟶wf 6329 ‘cfv 6333 (class class class)co 7164 1^st c1st 7705 2^nd c2nd 7706 ↑_pm cpm 8431 compcco 16673 dom_acdoma 17385 cod_accoda 17386 Arrowcarw 17387 Hom_achoma 17388 comp_accoa 17419

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1916 ax-6 1974 ax-7 2019 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2161 ax-12 2178 ax-ext 2710 ax-rep 5151 ax-sep 5164 ax-nul 5171 ax-pow 5229 ax-pr 5293 ax-un 7473

This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2074 df-mo 2540 df-eu 2570 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-ne 2935 df-ral 3058 df-rex 3059 df-reu 3060 df-rab 3062 df-v 3399 df-sbc 3680 df-csb 3789 df-dif 3844 df-un 3846 df-in 3848 df-ss 3858 df-nul 4210 df-if 4412 df-pw 4487 df-sn 4514 df-pr 4516 df-op 4520 df-ot 4522 df-uni 4794 df-iun 4880 df-br 5028 df-opab 5090 df-mpt 5108 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-iota 6291 df-fun 6335 df-fn 6336 df-f 6337 df-f1 6338 df-fo 6339 df-f1o 6340 df-fv 6341 df-ov 7167 df-oprab 7168 df-mpo 7169 df-1st 7707 df-2nd 7708 df-pm 8433 df-cat 17035 df-doma 17389 df-coda 17390 df-homa 17391 df-arw 17392 df-coa 17421

This theorem is referenced by: (None)

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