Theorem reldmran2 50416

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 5785	. . 3 ⊢ Rel ∅
2		df-ov 7413	. . . . . . 7 ⊢ (𝑃 Ran 𝐸) = ( Ran ‘⟨𝑃, 𝐸⟩)
3		id 23	. . . . . . 7 ⊢ (( Ran ‘⟨𝑃, 𝐸⟩) = ∅ → ( Ran ‘⟨𝑃, 𝐸⟩) = ∅)
4	2, 3	eqtrid 2810	. . . . . 6 ⊢ (( Ran ‘⟨𝑃, 𝐸⟩) = ∅ → (𝑃 Ran 𝐸) = ∅)
5	4	dmeqd 5895	. . . . 5 ⊢ (( Ran ‘⟨𝑃, 𝐸⟩) = ∅ → dom (𝑃 Ran 𝐸) = dom ∅)
6		dm0 5910	. . . . 5 ⊢ dom ∅ = ∅
7	5, 6	eqtrdi 2814	. . . 4 ⊢ (( Ran ‘⟨𝑃, 𝐸⟩) = ∅ → dom (𝑃 Ran 𝐸) = ∅)
8	7	releqd 5765	. . 3 ⊢ (( Ran ‘⟨𝑃, 𝐸⟩) = ∅ → (Rel dom (𝑃 Ran 𝐸) ↔ Rel ∅))
9	1, 8	mpbiri 261	. 2 ⊢ (( Ran ‘⟨𝑃, 𝐸⟩) = ∅ → Rel dom (𝑃 Ran 𝐸))
10		eqid 2763	. . . 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 7544	. . 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 6921	. . . . . . . . . 10 ⊢ (( Ran ‘⟨𝑃, 𝐸⟩) ≠ ∅ → (⟨𝑃, 𝐸⟩ ∈ dom Ran ∧ Fun ( Ran ↾ {⟨𝑃, 𝐸⟩})))
13	12	simpld 499	. . . . . . . . 9 ⊢ (( Ran ‘⟨𝑃, 𝐸⟩) ≠ ∅ → ⟨𝑃, 𝐸⟩ ∈ dom Ran )
14		ranfn 50408	. . . . . . . . . 10 ⊢ Ran Fn ((V × V) × V)
15	14	fndmi 6639	. . . . . . . . 9 ⊢ dom Ran = ((V × V) × V)
16	13, 15	eleqtrdi 2873	. . . . . . . 8 ⊢ (( Ran ‘⟨𝑃, 𝐸⟩) ≠ ∅ → ⟨𝑃, 𝐸⟩ ∈ ((V × V) × V))
17		opelxp1 5703	. . . . . . . 8 ⊢ (⟨𝑃, 𝐸⟩ ∈ ((V × V) × V) → 𝑃 ∈ (V × V))
18		1st2nd2 8021	. . . . . . . 8 ⊢ (𝑃 ∈ (V × V) → 𝑃 = ⟨(1st ‘𝑃), (2nd ‘𝑃)⟩)
19	16, 17, 18	3syl 19	. . . . . . 7 ⊢ (( Ran ‘⟨𝑃, 𝐸⟩) ≠ ∅ → 𝑃 = ⟨(1st ‘𝑃), (2nd ‘𝑃)⟩)
20	19	oveq1d 7425	. . . . . 6 ⊢ (( Ran ‘⟨𝑃, 𝐸⟩) ≠ ∅ → (𝑃 Ran 𝐸) = (⟨(1st ‘𝑃), (2nd ‘𝑃)⟩ Ran 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 6896	. . . . . . 7 ⊢ (( Ran ‘⟨𝑃, 𝐸⟩) ≠ ∅ → (1st ‘𝑃) ∈ V)
24		fvexd 6896	. . . . . . 7 ⊢ (( Ran ‘⟨𝑃, 𝐸⟩) ≠ ∅ → (2nd ‘𝑃) ∈ V)
25		opelxp2 5704	. . . . . . . 8 ⊢ (⟨𝑃, 𝐸⟩ ∈ ((V × V) × V) → 𝐸 ∈ V)
26	16, 25	syl 18	. . . . . . 7 ⊢ (( Ran ‘⟨𝑃, 𝐸⟩) ≠ ∅ → 𝐸 ∈ V)
27		eqid 2763	. . . . . . 7 ⊢ (oppCat‘((2nd ‘𝑃) FuncCat 𝐸)) = (oppCat‘((2nd ‘𝑃) FuncCat 𝐸))
28		eqid 2763	. . . . . . 7 ⊢ (oppCat‘((1st ‘𝑃) FuncCat 𝐸)) = (oppCat‘((1st ‘𝑃) FuncCat 𝐸))
29	21, 22, 23, 24, 26, 27, 28	ranfval 50412	. . . . . 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 2798	. . . . 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 5895	. . . 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 5765	. . 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 261	. 2 ⊢ (( Ran ‘⟨𝑃, 𝐸⟩) ≠ ∅ → Rel dom (𝑃 Ran 𝐸))
34	9, 33	pm2.61ine 3041	1 ⊢ Rel dom (𝑃 Ran 𝐸)

Syntax hints:   = wceq 1570   ∈ wcel 2143   ≠ wne 2958  Vcvv 3455  ∅c0 4286  {csn 4589  ⟨cop 4595   × cxp 5659  dom cdm 5661   ↾ cres 5663  Rel wrel 5666  Fun wfun 6530  ‘cfv 6536  (class class class)co 7410   ∈ cmpo 7412  1^st c1st 7980  2^nd c2nd 7981  oppCatcoppc 17762   Func cfunc 17906   FuncCat cfuc 17997   oppFunc coppf 49920   UP cup 49971   −∘_F cprcof 50171   Ran cran 50404

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-rep 5238  ax-sep 5257  ax-nul 5269  ax-pow 5336  ax-pr 5404  ax-un 7732

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-reu 3370  df-rab 3417  df-v 3457  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4488  df-pw 4564  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4873  df-iun 4958  df-br 5110  df-opab 5174  df-mpt 5193  df-id 5556  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-res 5673  df-ima 5674  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-ov 7413  df-oprab 7414  df-mpo 7415  df-1st 7982  df-2nd 7983  df-ran 50406

This theorem is referenced by: (None)