Theorem reldmran2 49597

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

Assertion

Ref	Expression
reldmran2	⊢ Rel dom (𝑃 Ran 𝐸)

Proof of Theorem reldmran2

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

Step	Hyp	Ref	Expression
1		rel0 5764	. . 3 ⊢ Rel ∅
2		df-ov 7392	. . . . . . 7 ⊢ (𝑃 Ran 𝐸) = ( Ran ‘⟨𝑃, 𝐸⟩)
3		id 22	. . . . . . 7 ⊢ (( Ran ‘⟨𝑃, 𝐸⟩) = ∅ → ( Ran ‘⟨𝑃, 𝐸⟩) = ∅)
4	2, 3	eqtrid 2777	. . . . . 6 ⊢ (( Ran ‘⟨𝑃, 𝐸⟩) = ∅ → (𝑃 Ran 𝐸) = ∅)
5	4	dmeqd 5871	. . . . 5 ⊢ (( Ran ‘⟨𝑃, 𝐸⟩) = ∅ → dom (𝑃 Ran 𝐸) = dom ∅)
6		dm0 5886	. . . . 5 ⊢ dom ∅ = ∅
7	5, 6	eqtrdi 2781	. . . 4 ⊢ (( Ran ‘⟨𝑃, 𝐸⟩) = ∅ → dom (𝑃 Ran 𝐸) = ∅)
8	7	releqd 5743	. . 3 ⊢ (( Ran ‘⟨𝑃, 𝐸⟩) = ∅ → (Rel dom (𝑃 Ran 𝐸) ↔ Rel ∅))
9	1, 8	mpbiri 258	. 2 ⊢ (( Ran ‘⟨𝑃, 𝐸⟩) = ∅ → Rel dom (𝑃 Ran 𝐸))
10		eqid 2730	. . . 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 7525	. . 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 6903	. . . . . . . . . 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 49589	. . . . . . . . . 10 ⊢ Ran Fn ((V × V) × V)
15	14	fndmi 6624	. . . . . . . . 9 ⊢ dom Ran = ((V × V) × V)
16	13, 15	eleqtrdi 2839	. . . . . . . 8 ⊢ (( Ran ‘⟨𝑃, 𝐸⟩) ≠ ∅ → ⟨𝑃, 𝐸⟩ ∈ ((V × V) × V))
17		opelxp1 5682	. . . . . . . 8 ⊢ (⟨𝑃, 𝐸⟩ ∈ ((V × V) × V) → 𝑃 ∈ (V × V))
18		1st2nd2 8009	. . . . . . . 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 7404	. . . . . 6 ⊢ (( Ran ‘⟨𝑃, 𝐸⟩) ≠ ∅ → (𝑃 Ran 𝐸) = (⟨(1st ‘𝑃), (2nd ‘𝑃)⟩ Ran 𝐸))
21		eqid 2730	. . . . . . 7 ⊢ ((2nd ‘𝑃) FuncCat 𝐸) = ((2nd ‘𝑃) FuncCat 𝐸)
22		eqid 2730	. . . . . . 7 ⊢ ((1st ‘𝑃) FuncCat 𝐸) = ((1st ‘𝑃) FuncCat 𝐸)
23		fvexd 6875	. . . . . . 7 ⊢ (( Ran ‘⟨𝑃, 𝐸⟩) ≠ ∅ → (1st ‘𝑃) ∈ V)
24		fvexd 6875	. . . . . . 7 ⊢ (( Ran ‘⟨𝑃, 𝐸⟩) ≠ ∅ → (2nd ‘𝑃) ∈ V)
25		opelxp2 5683	. . . . . . . 8 ⊢ (⟨𝑃, 𝐸⟩ ∈ ((V × V) × V) → 𝐸 ∈ V)
26	16, 25	syl 17	. . . . . . 7 ⊢ (( Ran ‘⟨𝑃, 𝐸⟩) ≠ ∅ → 𝐸 ∈ V)
27		eqid 2730	. . . . . . 7 ⊢ (oppCat‘((2nd ‘𝑃) FuncCat 𝐸)) = (oppCat‘((2nd ‘𝑃) FuncCat 𝐸))
28		eqid 2730	. . . . . . 7 ⊢ (oppCat‘((1st ‘𝑃) FuncCat 𝐸)) = (oppCat‘((1st ‘𝑃) FuncCat 𝐸))
29	21, 22, 23, 24, 26, 27, 28	ranfval 49593	. . . . . 6 ⊢ (( Ran ‘⟨𝑃, 𝐸⟩) ≠ ∅ → (⟨(1st ‘𝑃), (2nd ‘𝑃)⟩ Ran 𝐸) = (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 2765	. . . . 5 ⊢ (( Ran ‘⟨𝑃, 𝐸⟩) ≠ ∅ → (𝑃 Ran 𝐸) = (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 5871	. . . 4 ⊢ (( Ran ‘⟨𝑃, 𝐸⟩) ≠ ∅ → dom (𝑃 Ran 𝐸) = 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 5743	. . 3 ⊢ (( Ran ‘⟨𝑃, 𝐸⟩) ≠ ∅ → (Rel dom (𝑃 Ran 𝐸) ↔ 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 (𝑃 Ran 𝐸))
34	9, 33	pm2.61ine 3009	1 ⊢ Rel dom (𝑃 Ran 𝐸)

Syntax hints:   = wceq 1540   ∈ wcel 2109   ≠ wne 2926  Vcvv 3450  ∅c0 4298  {csn 4591  ⟨cop 4597   × cxp 5638  dom cdm 5640   ↾ cres 5642  Rel wrel 5645  Fun wfun 6507  ‘cfv 6513  (class class class)co 7389   ∈ cmpo 7391  1^st c1st 7968  2^nd c2nd 7969  oppCatcoppc 17678   Func cfunc 17822   FuncCat cfuc 17913   oppFunc coppf 49101   UP cup 49152   −∘_F cprcof 49352   Ran cran 49585

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 2702  ax-rep 5236  ax-sep 5253  ax-nul 5263  ax-pow 5322  ax-pr 5389  ax-un 7713

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 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-reu 3357  df-rab 3409  df-v 3452  df-sbc 3756  df-csb 3865  df-dif 3919  df-un 3921  df-in 3923  df-ss 3933  df-nul 4299  df-if 4491  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4874  df-iun 4959  df-br 5110  df-opab 5172  df-mpt 5191  df-id 5535  df-xp 5646  df-rel 5647  df-cnv 5648  df-co 5649  df-dm 5650  df-rn 5651  df-res 5652  df-ima 5653  df-iota 6466  df-fun 6515  df-fn 6516  df-f 6517  df-f1 6518  df-fo 6519  df-f1o 6520  df-fv 6521  df-ov 7392  df-oprab 7393  df-mpo 7394  df-1st 7970  df-2nd 7971  df-ran 49587

This theorem is referenced by: (None)