Theorem reldmlan2 51131

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 5785	. . 3 ⊢ Rel ∅
2		df-ov 7454	. . . . . . 7 ⊢ (𝑃 Lan 𝐸) = ( Lan ‘⟨𝑃, 𝐸⟩)
3		id 23	. . . . . . 7 ⊢ (( Lan ‘⟨𝑃, 𝐸⟩) = ∅ → ( Lan ‘⟨𝑃, 𝐸⟩) = ∅)
4	2, 3	eqtrid 2810	. . . . . 6 ⊢ (( Lan ‘⟨𝑃, 𝐸⟩) = ∅ → (𝑃 Lan 𝐸) = ∅)
5	4	dmeqd 5899	. . . . 5 ⊢ (( Lan ‘⟨𝑃, 𝐸⟩) = ∅ → dom (𝑃 Lan 𝐸) = dom ∅)
6		dm0 5914	. . . . 5 ⊢ dom ∅ = ∅
7	5, 6	eqtrdi 2814	. . . 4 ⊢ (( Lan ‘⟨𝑃, 𝐸⟩) = ∅ → dom (𝑃 Lan 𝐸) = ∅)
8	7	releqd 5763	. . 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 2763	. . . 4 ⊢ (𝑓 ∈ ((1st ‘𝑃) Func (2nd ‘𝑃)), 𝑥 ∈ ((1st ‘𝑃) Func 𝐸) ↦ ((⟨(2nd ‘𝑃), 𝐸⟩ −∘F 𝑓)(((2nd ‘𝑃) FuncCat 𝐸) UP ((1st ‘𝑃) FuncCat 𝐸))𝑥)) = (𝑓 ∈ ((1st ‘𝑃) Func (2nd ‘𝑃)), 𝑥 ∈ ((1st ‘𝑃) Func 𝐸) ↦ ((⟨(2nd ‘𝑃), 𝐸⟩ −∘F 𝑓)(((2nd ‘𝑃) FuncCat 𝐸) UP ((1st ‘𝑃) FuncCat 𝐸))𝑥))
11	10	reldmmpo 7584	. . 3 ⊢ Rel dom (𝑓 ∈ ((1st ‘𝑃) Func (2nd ‘𝑃)), 𝑥 ∈ ((1st ‘𝑃) Func 𝐸) ↦ ((⟨(2nd ‘𝑃), 𝐸⟩ −∘F 𝑓)(((2nd ‘𝑃) FuncCat 𝐸) UP ((1st ‘𝑃) FuncCat 𝐸))𝑥))
12		fvfundmfvn0 6956	. . . . . . . . . 10 ⊢ (( Lan ‘⟨𝑃, 𝐸⟩) ≠ ∅ → (⟨𝑃, 𝐸⟩ ∈ dom Lan ∧ Fun ( Lan ↾ {⟨𝑃, 𝐸⟩})))
13	12	simpld 500	. . . . . . . . 9 ⊢ (( Lan ‘⟨𝑃, 𝐸⟩) ≠ ∅ → ⟨𝑃, 𝐸⟩ ∈ dom Lan )
14		lanfn 51123	. . . . . . . . . 10 ⊢ Lan Fn ((V × V) × V)
15	14	fndmi 6673	. . . . . . . . 9 ⊢ dom Lan = ((V × V) × V)
16	13, 15	eleqtrdi 2874	. . . . . . . 8 ⊢ (( Lan ‘⟨𝑃, 𝐸⟩) ≠ ∅ → ⟨𝑃, 𝐸⟩ ∈ ((V × V) × V))
17		opelxp1 5700	. . . . . . . 8 ⊢ (⟨𝑃, 𝐸⟩ ∈ ((V × V) × V) → 𝑃 ∈ (V × V))
18		1st2nd2 8074	. . . . . . . 8 ⊢ (𝑃 ∈ (V × V) → 𝑃 = ⟨(1st ‘𝑃), (2nd ‘𝑃)⟩)
19	16, 17, 18	3syl 19	. . . . . . 7 ⊢ (( Lan ‘⟨𝑃, 𝐸⟩) ≠ ∅ → 𝑃 = ⟨(1st ‘𝑃), (2nd ‘𝑃)⟩)
20	19	oveq1d 7466	. . . . . 6 ⊢ (( Lan ‘⟨𝑃, 𝐸⟩) ≠ ∅ → (𝑃 Lan 𝐸) = (⟨(1st ‘𝑃), (2nd ‘𝑃)⟩ Lan 𝐸))
21		eqid 2763	. . . . . . 7 ⊢ ((2nd ‘𝑃) FuncCat 𝐸) = ((2nd ‘𝑃) FuncCat 𝐸)
22		eqid 2763	. . . . . . 7 ⊢ ((1st ‘𝑃) FuncCat 𝐸) = ((1st ‘𝑃) FuncCat 𝐸)
23		fvexd 6931	. . . . . . 7 ⊢ (( Lan ‘⟨𝑃, 𝐸⟩) ≠ ∅ → (1st ‘𝑃) ∈ V)
24		fvexd 6931	. . . . . . 7 ⊢ (( Lan ‘⟨𝑃, 𝐸⟩) ≠ ∅ → (2nd ‘𝑃) ∈ V)
25		opelxp2 5701	. . . . . . . 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 51127	. . . . . 6 ⊢ (( Lan ‘⟨𝑃, 𝐸⟩) ≠ ∅ → (⟨(1st ‘𝑃), (2nd ‘𝑃)⟩ Lan 𝐸) = (𝑓 ∈ ((1st ‘𝑃) Func (2nd ‘𝑃)), 𝑥 ∈ ((1st ‘𝑃) Func 𝐸) ↦ ((⟨(2nd ‘𝑃), 𝐸⟩ −∘F 𝑓)(((2nd ‘𝑃) FuncCat 𝐸) UP ((1st ‘𝑃) FuncCat 𝐸))𝑥)))
28	20, 27	eqtrd 2798	. . . . 5 ⊢ (( Lan ‘⟨𝑃, 𝐸⟩) ≠ ∅ → (𝑃 Lan 𝐸) = (𝑓 ∈ ((1st ‘𝑃) Func (2nd ‘𝑃)), 𝑥 ∈ ((1st ‘𝑃) Func 𝐸) ↦ ((⟨(2nd ‘𝑃), 𝐸⟩ −∘F 𝑓)(((2nd ‘𝑃) FuncCat 𝐸) UP ((1st ‘𝑃) FuncCat 𝐸))𝑥)))
29	28	dmeqd 5899	. . . 4 ⊢ (( Lan ‘⟨𝑃, 𝐸⟩) ≠ ∅ → dom (𝑃 Lan 𝐸) = dom (𝑓 ∈ ((1st ‘𝑃) Func (2nd ‘𝑃)), 𝑥 ∈ ((1st ‘𝑃) Func 𝐸) ↦ ((⟨(2nd ‘𝑃), 𝐸⟩ −∘F 𝑓)(((2nd ‘𝑃) FuncCat 𝐸) UP ((1st ‘𝑃) FuncCat 𝐸))𝑥)))
30	29	releqd 5763	. . 3 ⊢ (( Lan ‘⟨𝑃, 𝐸⟩) ≠ ∅ → (Rel dom (𝑃 Lan 𝐸) ↔ Rel dom (𝑓 ∈ ((1st ‘𝑃) Func (2nd ‘𝑃)), 𝑥 ∈ ((1st ‘𝑃) Func 𝐸) ↦ ((⟨(2nd ‘𝑃), 𝐸⟩ −∘F 𝑓)(((2nd ‘𝑃) FuncCat 𝐸) UP ((1st ‘𝑃) FuncCat 𝐸))𝑥))))
31	11, 30	mpbiri 261	. 2 ⊢ (( Lan ‘⟨𝑃, 𝐸⟩) ≠ ∅ → Rel dom (𝑃 Lan 𝐸))
32	9, 31	pm2.61ine 3042	1 ⊢ Rel dom (𝑃 Lan 𝐸)

This proof depends on syntax axioms:   = wceq 1570   ∈ wcel 2147   ≠ wne 2959  Vcvv 3454  ∅c0 4281  {csn 4588  ⟨cop 4840   × cxp 5656  dom cdm 5658   ↾ cres 5660  Rel wrel 5663  Fun wfun 6560  ‘cfv 6566  (class class class)co 7451   ∈ cmpo 7453  1^st c1st 8033  2^nd c2nd 8034   Func cfunc 18079   FuncCat cfuc 18170   UP cup 50620   −∘_F cprcof 50839   Lan clan 51119

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5238  ax-sep 5255  ax-nul 5266  ax-pow 5334  ax-pr 5398  ax-un 7783

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4841  df-uni 4873  df-iun 4958  df-br 5109  df-opab 5174  df-mpt 5193  df-id 5553  df-xp 5664  df-rel 5665  df-cnv 5666  df-co 5667  df-dm 5668  df-rn 5669  df-res 5670  df-ima 5671  df-iota 6520  df-fun 6568  df-fn 6569  df-f 6570  df-f1 6571  df-fo 6572  df-f1o 6573  df-fv 6574  df-ov 7454  df-oprab 7455  df-mpo 7456  df-1st 8035  df-2nd 8036  df-lan 51121

This theorem is used by: (None)