Theorem reldmran2 50455

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 5787	. . 3 ⊢ Rel ∅
2		df-ov 7422	. . . . . . 7 ⊢ (𝑃 Ran 𝐸) = ( Ran ‘⟨𝑃, 𝐸⟩)
3		id 23	. . . . . . 7 ⊢ (( Ran ‘⟨𝑃, 𝐸⟩) = ∅ → ( Ran ‘⟨𝑃, 𝐸⟩) = ∅)
4	2, 3	eqtrid 2812	. . . . . 6 ⊢ (( Ran ‘⟨𝑃, 𝐸⟩) = ∅ → (𝑃 Ran 𝐸) = ∅)
5	4	dmeqd 5897	. . . . 5 ⊢ (( Ran ‘⟨𝑃, 𝐸⟩) = ∅ → dom (𝑃 Ran 𝐸) = dom ∅)
6		dm0 5912	. . . . 5 ⊢ dom ∅ = ∅
7	5, 6	eqtrdi 2816	. . . 4 ⊢ (( Ran ‘⟨𝑃, 𝐸⟩) = ∅ → dom (𝑃 Ran 𝐸) = ∅)
8	7	releqd 5767	. . 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 2765	. . . 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 7553	. . 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 6925	. . . . . . . . . 10 ⊢ (( Ran ‘⟨𝑃, 𝐸⟩) ≠ ∅ → (⟨𝑃, 𝐸⟩ ∈ dom Ran ∧ Fun ( Ran ↾ {⟨𝑃, 𝐸⟩})))
13	12	simpld 500	. . . . . . . . 9 ⊢ (( Ran ‘⟨𝑃, 𝐸⟩) ≠ ∅ → ⟨𝑃, 𝐸⟩ ∈ dom Ran )
14		ranfn 50447	. . . . . . . . . 10 ⊢ Ran Fn ((V × V) × V)
15	14	fndmi 6643	. . . . . . . . 9 ⊢ dom Ran = ((V × V) × V)
16	13, 15	eleqtrdi 2875	. . . . . . . 8 ⊢ (( Ran ‘⟨𝑃, 𝐸⟩) ≠ ∅ → ⟨𝑃, 𝐸⟩ ∈ ((V × V) × V))
17		opelxp1 5705	. . . . . . . 8 ⊢ (⟨𝑃, 𝐸⟩ ∈ ((V × V) × V) → 𝑃 ∈ (V × V))
18		1st2nd2 8031	. . . . . . . 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 7434	. . . . . 6 ⊢ (( Ran ‘⟨𝑃, 𝐸⟩) ≠ ∅ → (𝑃 Ran 𝐸) = (⟨(1st ‘𝑃), (2nd ‘𝑃)⟩ Ran 𝐸))
21		eqid 2765	. . . . . . 7 ⊢ ((2nd ‘𝑃) FuncCat 𝐸) = ((2nd ‘𝑃) FuncCat 𝐸)
22		eqid 2765	. . . . . . 7 ⊢ ((1st ‘𝑃) FuncCat 𝐸) = ((1st ‘𝑃) FuncCat 𝐸)
23		fvexd 6900	. . . . . . 7 ⊢ (( Ran ‘⟨𝑃, 𝐸⟩) ≠ ∅ → (1st ‘𝑃) ∈ V)
24		fvexd 6900	. . . . . . 7 ⊢ (( Ran ‘⟨𝑃, 𝐸⟩) ≠ ∅ → (2nd ‘𝑃) ∈ V)
25		opelxp2 5706	. . . . . . . 8 ⊢ (⟨𝑃, 𝐸⟩ ∈ ((V × V) × V) → 𝐸 ∈ V)
26	16, 25	syl 18	. . . . . . 7 ⊢ (( Ran ‘⟨𝑃, 𝐸⟩) ≠ ∅ → 𝐸 ∈ V)
27		eqid 2765	. . . . . . 7 ⊢ (oppCat‘((2nd ‘𝑃) FuncCat 𝐸)) = (oppCat‘((2nd ‘𝑃) FuncCat 𝐸))
28		eqid 2765	. . . . . . 7 ⊢ (oppCat‘((1st ‘𝑃) FuncCat 𝐸)) = (oppCat‘((1st ‘𝑃) FuncCat 𝐸))
29	21, 22, 23, 24, 26, 27, 28	ranfval 50451	. . . . . 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 2800	. . . . 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 5897	. . . 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 5767	. . 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 3043	1 ⊢ Rel dom (𝑃 Ran 𝐸)

This proof depends on syntax axioms:   = wceq 1570   ∈ wcel 2146   ≠ wne 2960  Vcvv 3457  ∅c0 4286  {csn 4591  ⟨cop 4597   × cxp 5661  dom cdm 5663   ↾ cres 5665  Rel wrel 5668  Fun wfun 6534  ‘cfv 6540  (class class class)co 7419   ∈ cmpo 7421  1^st c1st 7990  2^nd c2nd 7991  oppCatcoppc 17791   Func cfunc 17935   FuncCat cfuc 18026   oppFunc coppf 49959   UP cup 50010   −∘_F cprcof 50210   Ran cran 50443

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 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737  ax-rep 5240  ax-sep 5259  ax-nul 5271  ax-pow 5338  ax-pr 5406  ax-un 7742

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 2100  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3082  df-rex 3092  df-reu 3372  df-rab 3419  df-v 3459  df-sbc 3747  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4287  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-iun 4960  df-br 5112  df-opab 5176  df-mpt 5195  df-id 5558  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-iota 6496  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-ov 7422  df-oprab 7423  df-mpo 7424  df-1st 7992  df-2nd 7993  df-ran 50445

This theorem is used by: (None)