reldmlan2 - Mathbox for Zhi Wang

	Mathbox for Zhi Wang		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > Mathboxes > reldmlan2		Structured version Visualization version GIF version

Theorem reldmlan2 49976

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 5756	. . 3 ⊢ Rel ∅
2		df-ov 7371	. . . . . . 7 ⊢ (𝑃 Lan 𝐸) = ( Lan ‘⟨𝑃, 𝐸⟩)
3		id 22	. . . . . . 7 ⊢ (( Lan ‘⟨𝑃, 𝐸⟩) = ∅ → ( Lan ‘⟨𝑃, 𝐸⟩) = ∅)
4	2, 3	eqtrid 2784	. . . . . 6 ⊢ (( Lan ‘⟨𝑃, 𝐸⟩) = ∅ → (𝑃 Lan 𝐸) = ∅)
5	4	dmeqd 5862	. . . . 5 ⊢ (( Lan ‘⟨𝑃, 𝐸⟩) = ∅ → dom (𝑃 Lan 𝐸) = dom ∅)
6		dm0 5877	. . . . 5 ⊢ dom ∅ = ∅
7	5, 6	eqtrdi 2788	. . . 4 ⊢ (( Lan ‘⟨𝑃, 𝐸⟩) = ∅ → dom (𝑃 Lan 𝐸) = ∅)
8	7	releqd 5736	. . 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 2737	. . . 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 7502	. . 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 6882	. . . . . . . . . 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 49968	. . . . . . . . . 10 ⊢ Lan Fn ((V × V) × V)
15	14	fndmi 6604	. . . . . . . . 9 ⊢ dom Lan = ((V × V) × V)
16	13, 15	eleqtrdi 2847	. . . . . . . 8 ⊢ (( Lan ‘⟨𝑃, 𝐸⟩) ≠ ∅ → ⟨𝑃, 𝐸⟩ ∈ ((V × V) × V))
17		opelxp1 5674	. . . . . . . 8 ⊢ (⟨𝑃, 𝐸⟩ ∈ ((V × V) × V) → 𝑃 ∈ (V × V))
18		1st2nd2 7982	. . . . . . . 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 7383	. . . . . 6 ⊢ (( Lan ‘⟨𝑃, 𝐸⟩) ≠ ∅ → (𝑃 Lan 𝐸) = (⟨(1^st ‘𝑃), (2^nd ‘𝑃)⟩ Lan 𝐸))
21		eqid 2737	. . . . . . 7 ⊢ ((2^nd ‘𝑃) FuncCat 𝐸) = ((2^nd ‘𝑃) FuncCat 𝐸)
22		eqid 2737	. . . . . . 7 ⊢ ((1^st ‘𝑃) FuncCat 𝐸) = ((1^st ‘𝑃) FuncCat 𝐸)
23		fvexd 6857	. . . . . . 7 ⊢ (( Lan ‘⟨𝑃, 𝐸⟩) ≠ ∅ → (1^st ‘𝑃) ∈ V)
24		fvexd 6857	. . . . . . 7 ⊢ (( Lan ‘⟨𝑃, 𝐸⟩) ≠ ∅ → (2^nd ‘𝑃) ∈ V)
25		opelxp2 5675	. . . . . . . 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 49972	. . . . . 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 2772	. . . . 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 5862	. . . 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 5736	. . 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 3016	1 ⊢ Rel dom (𝑃 Lan 𝐸)

Colors of variables: wff setvar class

Syntax hints: = wceq 1542 ∈ wcel 2114 ≠ wne 2933 Vcvv 3442 ∅c0 4287 {csn 4582 ⟨cop 4588 × cxp 5630 dom cdm 5632 ↾ cres 5634 Rel wrel 5637 Fun wfun 6494 ‘cfv 6500 (class class class)co 7368 ∈ cmpo 7370 1^st c1st 7941 2^nd c2nd 7942 Func cfunc 17790 FuncCat cfuc 17881 UP cup 49532 −∘_F cprcof 49732 Lan clan 49964

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-rep 5226 ax-sep 5243 ax-nul 5253 ax-pow 5312 ax-pr 5379 ax-un 7690

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-ral 3053 df-rex 3063 df-reu 3353 df-rab 3402 df-v 3444 df-sbc 3743 df-csb 3852 df-dif 3906 df-un 3908 df-in 3910 df-ss 3920 df-nul 4288 df-if 4482 df-pw 4558 df-sn 4583 df-pr 4585 df-op 4589 df-uni 4866 df-iun 4950 df-br 5101 df-opab 5163 df-mpt 5182 df-id 5527 df-xp 5638 df-rel 5639 df-cnv 5640 df-co 5641 df-dm 5642 df-rn 5643 df-res 5644 df-ima 5645 df-iota 6456 df-fun 6502 df-fn 6503 df-f 6504 df-f1 6505 df-fo 6506 df-f1o 6507 df-fv 6508 df-ov 7371 df-oprab 7372 df-mpo 7373 df-1st 7943 df-2nd 7944 df-lan 49966

This theorem is referenced by: (None)

W3C validator