Theorem reldmran2 49354

Description: The domain of (𝑃Ran𝐸) is a relation. (Contributed by Zhi Wang, 4-Nov-2025.)

Assertion

Ref	Expression
reldmran2	⊢ Rel dom (𝑃Ran𝐸)

Proof of Theorem reldmran2

Dummy variables 𝑓 𝑥 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		rel0 5776	. . 3 ⊢ Rel ∅
2		df-ov 7403	. . . . . . 7 ⊢ (𝑃Ran𝐸) = (Ran‘⟨𝑃, 𝐸⟩)
3		id 22	. . . . . . 7 ⊢ ((Ran‘⟨𝑃, 𝐸⟩) = ∅ → (Ran‘⟨𝑃, 𝐸⟩) = ∅)
4	2, 3	eqtrid 2781	. . . . . 6 ⊢ ((Ran‘⟨𝑃, 𝐸⟩) = ∅ → (𝑃Ran𝐸) = ∅)
5	4	dmeqd 5883	. . . . 5 ⊢ ((Ran‘⟨𝑃, 𝐸⟩) = ∅ → dom (𝑃Ran𝐸) = dom ∅)
6		dm0 5898	. . . . 5 ⊢ dom ∅ = ∅
7	5, 6	eqtrdi 2785	. . . 4 ⊢ ((Ran‘⟨𝑃, 𝐸⟩) = ∅ → dom (𝑃Ran𝐸) = ∅)
8	7	releqd 5755	. . 3 ⊢ ((Ran‘⟨𝑃, 𝐸⟩) = ∅ → (Rel dom (𝑃Ran𝐸) ↔ Rel ∅))
9	1, 8	mpbiri 258	. 2 ⊢ ((Ran‘⟨𝑃, 𝐸⟩) = ∅ → Rel dom (𝑃Ran𝐸))
10		eqid 2734	. . . 4 ⊢ (𝑓 ∈ ((1st ‘𝑃) Func (2nd ‘𝑃)), 𝑥 ∈ ((1st ‘𝑃) Func 𝐸) ↦ ((oppFunc‘(⟨(2nd ‘𝑃), 𝐸⟩ −∘F 𝑓))((oppCat‘((2nd ‘𝑃) FuncCat 𝐸))UP(oppCat‘((1st ‘𝑃) FuncCat 𝐸)))𝑥)) = (𝑓 ∈ ((1st ‘𝑃) Func (2nd ‘𝑃)), 𝑥 ∈ ((1st ‘𝑃) Func 𝐸) ↦ ((oppFunc‘(⟨(2nd ‘𝑃), 𝐸⟩ −∘F 𝑓))((oppCat‘((2nd ‘𝑃) FuncCat 𝐸))UP(oppCat‘((1st ‘𝑃) FuncCat 𝐸)))𝑥))
11	10	reldmmpo 7536	. . 3 ⊢ Rel dom (𝑓 ∈ ((1st ‘𝑃) Func (2nd ‘𝑃)), 𝑥 ∈ ((1st ‘𝑃) Func 𝐸) ↦ ((oppFunc‘(⟨(2nd ‘𝑃), 𝐸⟩ −∘F 𝑓))((oppCat‘((2nd ‘𝑃) FuncCat 𝐸))UP(oppCat‘((1st ‘𝑃) FuncCat 𝐸)))𝑥))
12		fvfundmfvn0 6916	. . . . . . . . . 10 ⊢ ((Ran‘⟨𝑃, 𝐸⟩) ≠ ∅ → (⟨𝑃, 𝐸⟩ ∈ dom Ran ∧ Fun (Ran ↾ {⟨𝑃, 𝐸⟩})))
13	12	simpld 494	. . . . . . . . 9 ⊢ ((Ran‘⟨𝑃, 𝐸⟩) ≠ ∅ → ⟨𝑃, 𝐸⟩ ∈ dom Ran)
14		ranfn 49348	. . . . . . . . . 10 ⊢ Ran Fn ((V × V) × V)
15	14	fndmi 6639	. . . . . . . . 9 ⊢ dom Ran = ((V × V) × V)
16	13, 15	eleqtrdi 2843	. . . . . . . 8 ⊢ ((Ran‘⟨𝑃, 𝐸⟩) ≠ ∅ → ⟨𝑃, 𝐸⟩ ∈ ((V × V) × V))
17		opelxp1 5694	. . . . . . . 8 ⊢ (⟨𝑃, 𝐸⟩ ∈ ((V × V) × V) → 𝑃 ∈ (V × V))
18		1st2nd2 8022	. . . . . . . 8 ⊢ (𝑃 ∈ (V × V) → 𝑃 = ⟨(1st ‘𝑃), (2nd ‘𝑃)⟩)
19	16, 17, 18	3syl 18	. . . . . . 7 ⊢ ((Ran‘⟨𝑃, 𝐸⟩) ≠ ∅ → 𝑃 = ⟨(1st ‘𝑃), (2nd ‘𝑃)⟩)
20	19	oveq1d 7415	. . . . . 6 ⊢ ((Ran‘⟨𝑃, 𝐸⟩) ≠ ∅ → (𝑃Ran𝐸) = (⟨(1st ‘𝑃), (2nd ‘𝑃)⟩Ran𝐸))
21		eqid 2734	. . . . . . 7 ⊢ ((2nd ‘𝑃) FuncCat 𝐸) = ((2nd ‘𝑃) FuncCat 𝐸)
22		eqid 2734	. . . . . . 7 ⊢ ((1st ‘𝑃) FuncCat 𝐸) = ((1st ‘𝑃) FuncCat 𝐸)
23		fvexd 6888	. . . . . . 7 ⊢ ((Ran‘⟨𝑃, 𝐸⟩) ≠ ∅ → (1st ‘𝑃) ∈ V)
24		fvexd 6888	. . . . . . 7 ⊢ ((Ran‘⟨𝑃, 𝐸⟩) ≠ ∅ → (2nd ‘𝑃) ∈ V)
25		opelxp2 5695	. . . . . . . 8 ⊢ (⟨𝑃, 𝐸⟩ ∈ ((V × V) × V) → 𝐸 ∈ V)
26	16, 25	syl 17	. . . . . . 7 ⊢ ((Ran‘⟨𝑃, 𝐸⟩) ≠ ∅ → 𝐸 ∈ V)
27		eqid 2734	. . . . . . 7 ⊢ (oppCat‘((2nd ‘𝑃) FuncCat 𝐸)) = (oppCat‘((2nd ‘𝑃) FuncCat 𝐸))
28		eqid 2734	. . . . . . 7 ⊢ (oppCat‘((1st ‘𝑃) FuncCat 𝐸)) = (oppCat‘((1st ‘𝑃) FuncCat 𝐸))
29	21, 22, 23, 24, 26, 27, 28	ranfval 49352	. . . . . 6 ⊢ ((Ran‘⟨𝑃, 𝐸⟩) ≠ ∅ → (⟨(1st ‘𝑃), (2nd ‘𝑃)⟩Ran𝐸) = (𝑓 ∈ ((1st ‘𝑃) Func (2nd ‘𝑃)), 𝑥 ∈ ((1st ‘𝑃) Func 𝐸) ↦ ((oppFunc‘(⟨(2nd ‘𝑃), 𝐸⟩ −∘F 𝑓))((oppCat‘((2nd ‘𝑃) FuncCat 𝐸))UP(oppCat‘((1st ‘𝑃) FuncCat 𝐸)))𝑥)))
30	20, 29	eqtrd 2769	. . . . 5 ⊢ ((Ran‘⟨𝑃, 𝐸⟩) ≠ ∅ → (𝑃Ran𝐸) = (𝑓 ∈ ((1st ‘𝑃) Func (2nd ‘𝑃)), 𝑥 ∈ ((1st ‘𝑃) Func 𝐸) ↦ ((oppFunc‘(⟨(2nd ‘𝑃), 𝐸⟩ −∘F 𝑓))((oppCat‘((2nd ‘𝑃) FuncCat 𝐸))UP(oppCat‘((1st ‘𝑃) FuncCat 𝐸)))𝑥)))
31	30	dmeqd 5883	. . . 4 ⊢ ((Ran‘⟨𝑃, 𝐸⟩) ≠ ∅ → dom (𝑃Ran𝐸) = dom (𝑓 ∈ ((1st ‘𝑃) Func (2nd ‘𝑃)), 𝑥 ∈ ((1st ‘𝑃) Func 𝐸) ↦ ((oppFunc‘(⟨(2nd ‘𝑃), 𝐸⟩ −∘F 𝑓))((oppCat‘((2nd ‘𝑃) FuncCat 𝐸))UP(oppCat‘((1st ‘𝑃) FuncCat 𝐸)))𝑥)))
32	31	releqd 5755	. . 3 ⊢ ((Ran‘⟨𝑃, 𝐸⟩) ≠ ∅ → (Rel dom (𝑃Ran𝐸) ↔ Rel dom (𝑓 ∈ ((1st ‘𝑃) Func (2nd ‘𝑃)), 𝑥 ∈ ((1st ‘𝑃) Func 𝐸) ↦ ((oppFunc‘(⟨(2nd ‘𝑃), 𝐸⟩ −∘F 𝑓))((oppCat‘((2nd ‘𝑃) FuncCat 𝐸))UP(oppCat‘((1st ‘𝑃) FuncCat 𝐸)))𝑥))))
33	11, 32	mpbiri 258	. 2 ⊢ ((Ran‘⟨𝑃, 𝐸⟩) ≠ ∅ → Rel dom (𝑃Ran𝐸))
34	9, 33	pm2.61ine 3014	1 ⊢ Rel dom (𝑃Ran𝐸)

Syntax hints:   = wceq 1539   ∈ wcel 2107   ≠ wne 2931  Vcvv 3457  ∅c0 4306  {csn 4599  ⟨cop 4605   × cxp 5650  dom cdm 5652   ↾ cres 5654  Rel wrel 5657  Fun wfun 6522  ‘cfv 6528  (class class class)co 7400   ∈ cmpo 7402  1^st c1st 7981  2^nd c2nd 7982  oppCatcoppc 17710   Func cfunc 17854   FuncCat cfuc 17945  oppFunccoppf 48950  UPcup 48974   −∘_F cprcof 49147  Rancran 49344

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2706  ax-rep 5247  ax-sep 5264  ax-nul 5274  ax-pow 5333  ax-pr 5400  ax-un 7724

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2808  df-nfc 2884  df-ne 2932  df-ral 3051  df-rex 3060  df-reu 3358  df-rab 3414  df-v 3459  df-sbc 3764  df-csb 3873  df-dif 3927  df-un 3929  df-in 3931  df-ss 3941  df-nul 4307  df-if 4499  df-pw 4575  df-sn 4600  df-pr 4602  df-op 4606  df-uni 4882  df-iun 4967  df-br 5118  df-opab 5180  df-mpt 5200  df-id 5546  df-xp 5658  df-rel 5659  df-cnv 5660  df-co 5661  df-dm 5662  df-rn 5663  df-res 5664  df-ima 5665  df-iota 6481  df-fun 6530  df-fn 6531  df-f 6532  df-f1 6533  df-fo 6534  df-f1o 6535  df-fv 6536  df-ov 7403  df-oprab 7404  df-mpo 7405  df-1st 7983  df-2nd 7984  df-ran 49346

This theorem is referenced by: (None)