reldmlan2 - Mathbox for Zhi Wang

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Theorem reldmlan2 49728

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 5738	. . 3 ⊢ Rel ∅
2		df-ov 7349	. . . . . . 7 ⊢ (𝑃 Lan 𝐸) = ( Lan ‘⟨𝑃, 𝐸⟩)
3		id 22	. . . . . . 7 ⊢ (( Lan ‘⟨𝑃, 𝐸⟩) = ∅ → ( Lan ‘⟨𝑃, 𝐸⟩) = ∅)
4	2, 3	eqtrid 2778	. . . . . 6 ⊢ (( Lan ‘⟨𝑃, 𝐸⟩) = ∅ → (𝑃 Lan 𝐸) = ∅)
5	4	dmeqd 5844	. . . . 5 ⊢ (( Lan ‘⟨𝑃, 𝐸⟩) = ∅ → dom (𝑃 Lan 𝐸) = dom ∅)
6		dm0 5859	. . . . 5 ⊢ dom ∅ = ∅
7	5, 6	eqtrdi 2782	. . . 4 ⊢ (( Lan ‘⟨𝑃, 𝐸⟩) = ∅ → dom (𝑃 Lan 𝐸) = ∅)
8	7	releqd 5718	. . 3 ⊢ (( Lan ‘⟨𝑃, 𝐸⟩) = ∅ → (Rel dom (𝑃 Lan 𝐸) ↔ Rel ∅))
9	1, 8	mpbiri 258	. 2 ⊢ (( Lan ‘⟨𝑃, 𝐸⟩) = ∅ → Rel dom (𝑃 Lan 𝐸))
10		eqid 2731	. . . 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 7480	. . 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 6862	. . . . . . . . . 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 49720	. . . . . . . . . 10 ⊢ Lan Fn ((V × V) × V)
15	14	fndmi 6585	. . . . . . . . 9 ⊢ dom Lan = ((V × V) × V)
16	13, 15	eleqtrdi 2841	. . . . . . . 8 ⊢ (( Lan ‘⟨𝑃, 𝐸⟩) ≠ ∅ → ⟨𝑃, 𝐸⟩ ∈ ((V × V) × V))
17		opelxp1 5656	. . . . . . . 8 ⊢ (⟨𝑃, 𝐸⟩ ∈ ((V × V) × V) → 𝑃 ∈ (V × V))
18		1st2nd2 7960	. . . . . . . 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 7361	. . . . . 6 ⊢ (( Lan ‘⟨𝑃, 𝐸⟩) ≠ ∅ → (𝑃 Lan 𝐸) = (⟨(1^st ‘𝑃), (2^nd ‘𝑃)⟩ Lan 𝐸))
21		eqid 2731	. . . . . . 7 ⊢ ((2^nd ‘𝑃) FuncCat 𝐸) = ((2^nd ‘𝑃) FuncCat 𝐸)
22		eqid 2731	. . . . . . 7 ⊢ ((1^st ‘𝑃) FuncCat 𝐸) = ((1^st ‘𝑃) FuncCat 𝐸)
23		fvexd 6837	. . . . . . 7 ⊢ (( Lan ‘⟨𝑃, 𝐸⟩) ≠ ∅ → (1^st ‘𝑃) ∈ V)
24		fvexd 6837	. . . . . . 7 ⊢ (( Lan ‘⟨𝑃, 𝐸⟩) ≠ ∅ → (2^nd ‘𝑃) ∈ V)
25		opelxp2 5657	. . . . . . . 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 49724	. . . . . 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 2766	. . . . 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 5844	. . . 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 5718	. . 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 258	. 2 ⊢ (( Lan ‘⟨𝑃, 𝐸⟩) ≠ ∅ → Rel dom (𝑃 Lan 𝐸))
32	9, 31	pm2.61ine 3011	1 ⊢ Rel dom (𝑃 Lan 𝐸)

Colors of variables: wff setvar class

Syntax hints: = wceq 1541 ∈ wcel 2111 ≠ wne 2928 Vcvv 3436 ∅c0 4280 {csn 4573 ⟨cop 4579 × cxp 5612 dom cdm 5614 ↾ cres 5616 Rel wrel 5619 Fun wfun 6475 ‘cfv 6481 (class class class)co 7346 ∈ cmpo 7348 1^st c1st 7919 2^nd c2nd 7920 Func cfunc 17761 FuncCat cfuc 17852 UP cup 49284 −∘_F cprcof 49484 Lan clan 49716

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2113 ax-9 2121 ax-10 2144 ax-11 2160 ax-12 2180 ax-ext 2703 ax-rep 5215 ax-sep 5232 ax-nul 5242 ax-pow 5301 ax-pr 5368 ax-un 7668

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2535 df-eu 2564 df-clab 2710 df-cleq 2723 df-clel 2806 df-nfc 2881 df-ne 2929 df-ral 3048 df-rex 3057 df-reu 3347 df-rab 3396 df-v 3438 df-sbc 3737 df-csb 3846 df-dif 3900 df-un 3902 df-in 3904 df-ss 3914 df-nul 4281 df-if 4473 df-pw 4549 df-sn 4574 df-pr 4576 df-op 4580 df-uni 4857 df-iun 4941 df-br 5090 df-opab 5152 df-mpt 5171 df-id 5509 df-xp 5620 df-rel 5621 df-cnv 5622 df-co 5623 df-dm 5624 df-rn 5625 df-res 5626 df-ima 5627 df-iota 6437 df-fun 6483 df-fn 6484 df-f 6485 df-f1 6486 df-fo 6487 df-f1o 6488 df-fv 6489 df-ov 7349 df-oprab 7350 df-mpo 7351 df-1st 7921 df-2nd 7922 df-lan 49718

This theorem is referenced by: (None)

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