reldmlan2 - Mathbox for Zhi Wang

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Theorem reldmlan2 49353

Description: The domain of (𝑃Lan𝐸) is a relation. (Contributed by Zhi Wang, 3-Nov-2025.)

**Assertion**
Ref	Expression
reldmlan2	⊢ Rel dom (𝑃Lan𝐸)

Proof of Theorem reldmlan2 Dummy variables 𝑓 𝑥 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		rel0 5776	. . 3 ⊢ Rel ∅
2		df-ov 7403	. . . . . . 7 ⊢ (𝑃Lan𝐸) = (Lan‘⟨𝑃, 𝐸⟩)
3		id 22	. . . . . . 7 ⊢ ((Lan‘⟨𝑃, 𝐸⟩) = ∅ → (Lan‘⟨𝑃, 𝐸⟩) = ∅)
4	2, 3	eqtrid 2781	. . . . . 6 ⊢ ((Lan‘⟨𝑃, 𝐸⟩) = ∅ → (𝑃Lan𝐸) = ∅)
5	4	dmeqd 5883	. . . . 5 ⊢ ((Lan‘⟨𝑃, 𝐸⟩) = ∅ → dom (𝑃Lan𝐸) = dom ∅)
6		dm0 5898	. . . . 5 ⊢ dom ∅ = ∅
7	5, 6	eqtrdi 2785	. . . 4 ⊢ ((Lan‘⟨𝑃, 𝐸⟩) = ∅ → dom (𝑃Lan𝐸) = ∅)
8	7	releqd 5755	. . 3 ⊢ ((Lan‘⟨𝑃, 𝐸⟩) = ∅ → (Rel dom (𝑃Lan𝐸) ↔ Rel ∅))
9	1, 8	mpbiri 258	. 2 ⊢ ((Lan‘⟨𝑃, 𝐸⟩) = ∅ → Rel dom (𝑃Lan𝐸))
10		eqid 2734	. . . 4 ⊢ (𝑓 ∈ ((1^st ‘𝑃) Func (2^nd ‘𝑃)), 𝑥 ∈ ((1^st ‘𝑃) Func 𝐸) ↦ ((⟨(2^nd ‘𝑃), 𝐸⟩ −∘_F 𝑓)(((2^nd ‘𝑃) FuncCat 𝐸)UP((1^st ‘𝑃) FuncCat 𝐸))𝑥)) = (𝑓 ∈ ((1^st ‘𝑃) Func (2^nd ‘𝑃)), 𝑥 ∈ ((1^st ‘𝑃) Func 𝐸) ↦ ((⟨(2^nd ‘𝑃), 𝐸⟩ −∘_F 𝑓)(((2^nd ‘𝑃) FuncCat 𝐸)UP((1^st ‘𝑃) FuncCat 𝐸))𝑥))
11	10	reldmmpo 7536	. . 3 ⊢ Rel dom (𝑓 ∈ ((1^st ‘𝑃) Func (2^nd ‘𝑃)), 𝑥 ∈ ((1^st ‘𝑃) Func 𝐸) ↦ ((⟨(2^nd ‘𝑃), 𝐸⟩ −∘_F 𝑓)(((2^nd ‘𝑃) FuncCat 𝐸)UP((1^st ‘𝑃) FuncCat 𝐸))𝑥))
12		fvfundmfvn0 6916	. . . . . . . . . 10 ⊢ ((Lan‘⟨𝑃, 𝐸⟩) ≠ ∅ → (⟨𝑃, 𝐸⟩ ∈ dom Lan ∧ Fun (Lan ↾ {⟨𝑃, 𝐸⟩})))
13	12	simpld 494	. . . . . . . . 9 ⊢ ((Lan‘⟨𝑃, 𝐸⟩) ≠ ∅ → ⟨𝑃, 𝐸⟩ ∈ dom Lan)
14		lanfn 49347	. . . . . . . . . 10 ⊢ Lan Fn ((V × V) × V)
15	14	fndmi 6639	. . . . . . . . 9 ⊢ dom Lan = ((V × V) × V)
16	13, 15	eleqtrdi 2843	. . . . . . . 8 ⊢ ((Lan‘⟨𝑃, 𝐸⟩) ≠ ∅ → ⟨𝑃, 𝐸⟩ ∈ ((V × V) × V))
17		opelxp1 5694	. . . . . . . 8 ⊢ (⟨𝑃, 𝐸⟩ ∈ ((V × V) × V) → 𝑃 ∈ (V × V))
18		1st2nd2 8022	. . . . . . . 8 ⊢ (𝑃 ∈ (V × V) → 𝑃 = ⟨(1^st ‘𝑃), (2^nd ‘𝑃)⟩)
19	16, 17, 18	3syl 18	. . . . . . 7 ⊢ ((Lan‘⟨𝑃, 𝐸⟩) ≠ ∅ → 𝑃 = ⟨(1^st ‘𝑃), (2^nd ‘𝑃)⟩)
20	19	oveq1d 7415	. . . . . 6 ⊢ ((Lan‘⟨𝑃, 𝐸⟩) ≠ ∅ → (𝑃Lan𝐸) = (⟨(1^st ‘𝑃), (2^nd ‘𝑃)⟩Lan𝐸))
21		eqid 2734	. . . . . . 7 ⊢ ((2^nd ‘𝑃) FuncCat 𝐸) = ((2^nd ‘𝑃) FuncCat 𝐸)
22		eqid 2734	. . . . . . 7 ⊢ ((1^st ‘𝑃) FuncCat 𝐸) = ((1^st ‘𝑃) FuncCat 𝐸)
23		fvexd 6888	. . . . . . 7 ⊢ ((Lan‘⟨𝑃, 𝐸⟩) ≠ ∅ → (1^st ‘𝑃) ∈ V)
24		fvexd 6888	. . . . . . 7 ⊢ ((Lan‘⟨𝑃, 𝐸⟩) ≠ ∅ → (2^nd ‘𝑃) ∈ V)
25		opelxp2 5695	. . . . . . . 8 ⊢ (⟨𝑃, 𝐸⟩ ∈ ((V × V) × V) → 𝐸 ∈ V)
26	16, 25	syl 17	. . . . . . 7 ⊢ ((Lan‘⟨𝑃, 𝐸⟩) ≠ ∅ → 𝐸 ∈ V)
27	21, 22, 23, 24, 26	lanfval 49351	. . . . . 6 ⊢ ((Lan‘⟨𝑃, 𝐸⟩) ≠ ∅ → (⟨(1^st ‘𝑃), (2^nd ‘𝑃)⟩Lan𝐸) = (𝑓 ∈ ((1^st ‘𝑃) Func (2^nd ‘𝑃)), 𝑥 ∈ ((1^st ‘𝑃) Func 𝐸) ↦ ((⟨(2^nd ‘𝑃), 𝐸⟩ −∘_F 𝑓)(((2^nd ‘𝑃) FuncCat 𝐸)UP((1^st ‘𝑃) FuncCat 𝐸))𝑥)))
28	20, 27	eqtrd 2769	. . . . 5 ⊢ ((Lan‘⟨𝑃, 𝐸⟩) ≠ ∅ → (𝑃Lan𝐸) = (𝑓 ∈ ((1^st ‘𝑃) Func (2^nd ‘𝑃)), 𝑥 ∈ ((1^st ‘𝑃) Func 𝐸) ↦ ((⟨(2^nd ‘𝑃), 𝐸⟩ −∘_F 𝑓)(((2^nd ‘𝑃) FuncCat 𝐸)UP((1^st ‘𝑃) FuncCat 𝐸))𝑥)))
29	28	dmeqd 5883	. . . 4 ⊢ ((Lan‘⟨𝑃, 𝐸⟩) ≠ ∅ → dom (𝑃Lan𝐸) = dom (𝑓 ∈ ((1^st ‘𝑃) Func (2^nd ‘𝑃)), 𝑥 ∈ ((1^st ‘𝑃) Func 𝐸) ↦ ((⟨(2^nd ‘𝑃), 𝐸⟩ −∘_F 𝑓)(((2^nd ‘𝑃) FuncCat 𝐸)UP((1^st ‘𝑃) FuncCat 𝐸))𝑥)))
30	29	releqd 5755	. . 3 ⊢ ((Lan‘⟨𝑃, 𝐸⟩) ≠ ∅ → (Rel dom (𝑃Lan𝐸) ↔ Rel dom (𝑓 ∈ ((1^st ‘𝑃) Func (2^nd ‘𝑃)), 𝑥 ∈ ((1^st ‘𝑃) Func 𝐸) ↦ ((⟨(2^nd ‘𝑃), 𝐸⟩ −∘_F 𝑓)(((2^nd ‘𝑃) FuncCat 𝐸)UP((1^st ‘𝑃) FuncCat 𝐸))𝑥))))
31	11, 30	mpbiri 258	. 2 ⊢ ((Lan‘⟨𝑃, 𝐸⟩) ≠ ∅ → Rel dom (𝑃Lan𝐸))
32	9, 31	pm2.61ine 3014	1 ⊢ Rel dom (𝑃Lan𝐸)

Colors of variables: wff setvar class

Syntax hints: = wceq 1539 ∈ wcel 2107 ≠ wne 2931 Vcvv 3457 ∅c0 4306 {csn 4599 ⟨cop 4605 × cxp 5650 dom cdm 5652 ↾ cres 5654 Rel wrel 5657 Fun wfun 6522 ‘cfv 6528 (class class class)co 7400 ∈ cmpo 7402 1^st c1st 7981 2^nd c2nd 7982 Func cfunc 17854 FuncCat cfuc 17945 UPcup 48974 −∘_F cprcof 49147 Lanclan 49343

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2706 ax-rep 5247 ax-sep 5264 ax-nul 5274 ax-pow 5333 ax-pr 5400 ax-un 7724

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2726 df-clel 2808 df-nfc 2884 df-ne 2932 df-ral 3051 df-rex 3060 df-reu 3358 df-rab 3414 df-v 3459 df-sbc 3764 df-csb 3873 df-dif 3927 df-un 3929 df-in 3931 df-ss 3941 df-nul 4307 df-if 4499 df-pw 4575 df-sn 4600 df-pr 4602 df-op 4606 df-uni 4882 df-iun 4967 df-br 5118 df-opab 5180 df-mpt 5200 df-id 5546 df-xp 5658 df-rel 5659 df-cnv 5660 df-co 5661 df-dm 5662 df-rn 5663 df-res 5664 df-ima 5665 df-iota 6481 df-fun 6530 df-fn 6531 df-f 6532 df-f1 6533 df-fo 6534 df-f1o 6535 df-fv 6536 df-ov 7403 df-oprab 7404 df-mpo 7405 df-1st 7983 df-2nd 7984 df-lan 49345

This theorem is referenced by: (None)

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