reldmlan2 - Mathbox for Zhi Wang

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Theorem reldmlan2 50280

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

**Assertion**
Ref	Expression
reldmlan2	⊢ Rel dom (𝑃 Lan 𝐸)

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

Step	Hyp	Ref	Expression
1		rel0 5786	. . 3 ⊢ Rel ∅
2		df-ov 7414	. . . . . . 7 ⊢ (𝑃 Lan 𝐸) = ( Lan ‘⟨𝑃, 𝐸⟩)
3		id 23	. . . . . . 7 ⊢ (( Lan ‘⟨𝑃, 𝐸⟩) = ∅ → ( Lan ‘⟨𝑃, 𝐸⟩) = ∅)
4	2, 3	eqtrid 2816	. . . . . 6 ⊢ (( Lan ‘⟨𝑃, 𝐸⟩) = ∅ → (𝑃 Lan 𝐸) = ∅)
5	4	dmeqd 5896	. . . . 5 ⊢ (( Lan ‘⟨𝑃, 𝐸⟩) = ∅ → dom (𝑃 Lan 𝐸) = dom ∅)
6		dm0 5911	. . . . 5 ⊢ dom ∅ = ∅
7	5, 6	eqtrdi 2820	. . . 4 ⊢ (( Lan ‘⟨𝑃, 𝐸⟩) = ∅ → dom (𝑃 Lan 𝐸) = ∅)
8	7	releqd 5766	. . 3 ⊢ (( Lan ‘⟨𝑃, 𝐸⟩) = ∅ → (Rel dom (𝑃 Lan 𝐸) ↔ Rel ∅))
9	1, 8	mpbiri 261	. 2 ⊢ (( Lan ‘⟨𝑃, 𝐸⟩) = ∅ → Rel dom (𝑃 Lan 𝐸))
10		eqid 2769	. . . 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 7545	. . 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 6922	. . . . . . . . . 10 ⊢ (( Lan ‘⟨𝑃, 𝐸⟩) ≠ ∅ → (⟨𝑃, 𝐸⟩ ∈ dom Lan ∧ Fun ( Lan ↾ {⟨𝑃, 𝐸⟩})))
13	12	simpld 499	. . . . . . . . 9 ⊢ (( Lan ‘⟨𝑃, 𝐸⟩) ≠ ∅ → ⟨𝑃, 𝐸⟩ ∈ dom Lan )
14		lanfn 50272	. . . . . . . . . 10 ⊢ Lan Fn ((V × V) × V)
15	14	fndmi 6640	. . . . . . . . 9 ⊢ dom Lan = ((V × V) × V)
16	13, 15	eleqtrdi 2879	. . . . . . . 8 ⊢ (( Lan ‘⟨𝑃, 𝐸⟩) ≠ ∅ → ⟨𝑃, 𝐸⟩ ∈ ((V × V) × V))
17		opelxp1 5704	. . . . . . . 8 ⊢ (⟨𝑃, 𝐸⟩ ∈ ((V × V) × V) → 𝑃 ∈ (V × V))
18		1st2nd2 8025	. . . . . . . 8 ⊢ (𝑃 ∈ (V × V) → 𝑃 = ⟨(1^st ‘𝑃), (2^nd ‘𝑃)⟩)
19	16, 17, 18	3syl 19	. . . . . . 7 ⊢ (( Lan ‘⟨𝑃, 𝐸⟩) ≠ ∅ → 𝑃 = ⟨(1^st ‘𝑃), (2^nd ‘𝑃)⟩)
20	19	oveq1d 7426	. . . . . 6 ⊢ (( Lan ‘⟨𝑃, 𝐸⟩) ≠ ∅ → (𝑃 Lan 𝐸) = (⟨(1^st ‘𝑃), (2^nd ‘𝑃)⟩ Lan 𝐸))
21		eqid 2769	. . . . . . 7 ⊢ ((2^nd ‘𝑃) FuncCat 𝐸) = ((2^nd ‘𝑃) FuncCat 𝐸)
22		eqid 2769	. . . . . . 7 ⊢ ((1^st ‘𝑃) FuncCat 𝐸) = ((1^st ‘𝑃) FuncCat 𝐸)
23		fvexd 6897	. . . . . . 7 ⊢ (( Lan ‘⟨𝑃, 𝐸⟩) ≠ ∅ → (1^st ‘𝑃) ∈ V)
24		fvexd 6897	. . . . . . 7 ⊢ (( Lan ‘⟨𝑃, 𝐸⟩) ≠ ∅ → (2^nd ‘𝑃) ∈ V)
25		opelxp2 5705	. . . . . . . 8 ⊢ (⟨𝑃, 𝐸⟩ ∈ ((V × V) × V) → 𝐸 ∈ V)
26	16, 25	syl 18	. . . . . . 7 ⊢ (( Lan ‘⟨𝑃, 𝐸⟩) ≠ ∅ → 𝐸 ∈ V)
27	21, 22, 23, 24, 26	lanfval 50276	. . . . . 6 ⊢ (( Lan ‘⟨𝑃, 𝐸⟩) ≠ ∅ → (⟨(1^st ‘𝑃), (2^nd ‘𝑃)⟩ Lan 𝐸) = (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 2804	. . . . 5 ⊢ (( Lan ‘⟨𝑃, 𝐸⟩) ≠ ∅ → (𝑃 Lan 𝐸) = (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 5896	. . . 4 ⊢ (( Lan ‘⟨𝑃, 𝐸⟩) ≠ ∅ → dom (𝑃 Lan 𝐸) = 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 5766	. . 3 ⊢ (( Lan ‘⟨𝑃, 𝐸⟩) ≠ ∅ → (Rel dom (𝑃 Lan 𝐸) ↔ 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 261	. 2 ⊢ (( Lan ‘⟨𝑃, 𝐸⟩) ≠ ∅ → Rel dom (𝑃 Lan 𝐸))
32	9, 31	pm2.61ine 3047	1 ⊢ Rel dom (𝑃 Lan 𝐸)

Colors of variables: wff setvar class

Syntax hints: = wceq 1567 ∈ wcel 2149 ≠ wne 2964 Vcvv 3463 ∅c0 4294 {csn 4594 ⟨cop 4600 × cxp 5660 dom cdm 5662 ↾ cres 5664 Rel wrel 5667 Fun wfun 6531 ‘cfv 6537 (class class class)co 7411 ∈ cmpo 7413 1^st c1st 7984 2^nd c2nd 7985 Func cfunc 17911 FuncCat cfuc 18002 UP cup 49836 −∘_F cprcof 50036 Lan clan 50268

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1822 ax-4 1836 ax-5 1937 ax-6 1994 ax-7 2035 ax-8 2151 ax-9 2159 ax-10 2182 ax-11 2198 ax-12 2219 ax-ext 2741 ax-rep 5242 ax-sep 5261 ax-nul 5271 ax-pow 5337 ax-pr 5405 ax-un 7733

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3an 1103 df-tru 1570 df-fal 1580 df-ex 1807 df-nf 1811 df-sb 2098 df-mo 2573 df-eu 2603 df-clab 2748 df-cleq 2761 df-clel 2844 df-nfc 2918 df-ne 2965 df-ral 3086 df-rex 3096 df-reu 3377 df-rab 3424 df-v 3465 df-sbc 3754 df-csb 3862 df-dif 3916 df-un 3918 df-in 3920 df-ss 3930 df-nul 4295 df-if 4493 df-pw 4569 df-sn 4595 df-pr 4597 df-op 4601 df-uni 4877 df-iun 4962 df-br 5114 df-opab 5178 df-mpt 5197 df-id 5557 df-xp 5668 df-rel 5669 df-cnv 5670 df-co 5671 df-dm 5672 df-rn 5673 df-res 5674 df-ima 5675 df-iota 6493 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-ov 7414 df-oprab 7415 df-mpo 7416 df-1st 7986 df-2nd 7987 df-lan 50270

This theorem is referenced by: (None)

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