reldmlan2 - Mathbox for Zhi Wang

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Theorem reldmlan2 49616

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 5736	. . 3 ⊢ Rel ∅
2		df-ov 7343	. . . . . . 7 ⊢ (𝑃 Lan 𝐸) = ( Lan ‘⟨𝑃, 𝐸⟩)
3		id 22	. . . . . . 7 ⊢ (( Lan ‘⟨𝑃, 𝐸⟩) = ∅ → ( Lan ‘⟨𝑃, 𝐸⟩) = ∅)
4	2, 3	eqtrid 2776	. . . . . 6 ⊢ (( Lan ‘⟨𝑃, 𝐸⟩) = ∅ → (𝑃 Lan 𝐸) = ∅)
5	4	dmeqd 5842	. . . . 5 ⊢ (( Lan ‘⟨𝑃, 𝐸⟩) = ∅ → dom (𝑃 Lan 𝐸) = dom ∅)
6		dm0 5857	. . . . 5 ⊢ dom ∅ = ∅
7	5, 6	eqtrdi 2780	. . . 4 ⊢ (( Lan ‘⟨𝑃, 𝐸⟩) = ∅ → dom (𝑃 Lan 𝐸) = ∅)
8	7	releqd 5716	. . 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 2729	. . . 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 7474	. . 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 6856	. . . . . . . . . 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 49608	. . . . . . . . . 10 ⊢ Lan Fn ((V × V) × V)
15	14	fndmi 6580	. . . . . . . . 9 ⊢ dom Lan = ((V × V) × V)
16	13, 15	eleqtrdi 2838	. . . . . . . 8 ⊢ (( Lan ‘⟨𝑃, 𝐸⟩) ≠ ∅ → ⟨𝑃, 𝐸⟩ ∈ ((V × V) × V))
17		opelxp1 5655	. . . . . . . 8 ⊢ (⟨𝑃, 𝐸⟩ ∈ ((V × V) × V) → 𝑃 ∈ (V × V))
18		1st2nd2 7954	. . . . . . . 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 7355	. . . . . 6 ⊢ (( Lan ‘⟨𝑃, 𝐸⟩) ≠ ∅ → (𝑃 Lan 𝐸) = (⟨(1^st ‘𝑃), (2^nd ‘𝑃)⟩ Lan 𝐸))
21		eqid 2729	. . . . . . 7 ⊢ ((2^nd ‘𝑃) FuncCat 𝐸) = ((2^nd ‘𝑃) FuncCat 𝐸)
22		eqid 2729	. . . . . . 7 ⊢ ((1^st ‘𝑃) FuncCat 𝐸) = ((1^st ‘𝑃) FuncCat 𝐸)
23		fvexd 6831	. . . . . . 7 ⊢ (( Lan ‘⟨𝑃, 𝐸⟩) ≠ ∅ → (1^st ‘𝑃) ∈ V)
24		fvexd 6831	. . . . . . 7 ⊢ (( Lan ‘⟨𝑃, 𝐸⟩) ≠ ∅ → (2^nd ‘𝑃) ∈ V)
25		opelxp2 5656	. . . . . . . 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 49612	. . . . . 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 2764	. . . . 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 5842	. . . 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 5716	. . 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 3008	1 ⊢ Rel dom (𝑃 Lan 𝐸)

Colors of variables: wff setvar class

Syntax hints: = wceq 1540 ∈ wcel 2109 ≠ wne 2925 Vcvv 3433 ∅c0 4280 {csn 4573 ⟨cop 4579 × cxp 5611 dom cdm 5613 ↾ cres 5615 Rel wrel 5618 Fun wfun 6470 ‘cfv 6476 (class class class)co 7340 ∈ cmpo 7342 1^st c1st 7913 2^nd c2nd 7914 Func cfunc 17748 FuncCat cfuc 17839 UP cup 49172 −∘_F cprcof 49372 Lan clan 49604

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-rep 5214 ax-sep 5231 ax-nul 5241 ax-pow 5300 ax-pr 5367 ax-un 7662

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-ral 3045 df-rex 3054 df-reu 3344 df-rab 3393 df-v 3435 df-sbc 3739 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-nul 4281 df-if 4473 df-pw 4549 df-sn 4574 df-pr 4576 df-op 4580 df-uni 4857 df-iun 4940 df-br 5089 df-opab 5151 df-mpt 5170 df-id 5508 df-xp 5619 df-rel 5620 df-cnv 5621 df-co 5622 df-dm 5623 df-rn 5624 df-res 5625 df-ima 5626 df-iota 6432 df-fun 6478 df-fn 6479 df-f 6480 df-f1 6481 df-fo 6482 df-f1o 6483 df-fv 6484 df-ov 7343 df-oprab 7344 df-mpo 7345 df-1st 7915 df-2nd 7916 df-lan 49606

This theorem is referenced by: (None)

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