Theorem reldmran2 50108

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 5749	. . 3 ⊢ Rel ∅
2		df-ov 7364	. . . . . . 7 ⊢ (𝑃 Ran 𝐸) = ( Ran ‘⟨𝑃, 𝐸⟩)
3		id 22	. . . . . . 7 ⊢ (( Ran ‘⟨𝑃, 𝐸⟩) = ∅ → ( Ran ‘⟨𝑃, 𝐸⟩) = ∅)
4	2, 3	eqtrid 2784	. . . . . 6 ⊢ (( Ran ‘⟨𝑃, 𝐸⟩) = ∅ → (𝑃 Ran 𝐸) = ∅)
5	4	dmeqd 5855	. . . . 5 ⊢ (( Ran ‘⟨𝑃, 𝐸⟩) = ∅ → dom (𝑃 Ran 𝐸) = dom ∅)
6		dm0 5870	. . . . 5 ⊢ dom ∅ = ∅
7	5, 6	eqtrdi 2788	. . . 4 ⊢ (( Ran ‘⟨𝑃, 𝐸⟩) = ∅ → dom (𝑃 Ran 𝐸) = ∅)
8	7	releqd 5729	. . 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 2737	. . . 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 7495	. . 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 6875	. . . . . . . . . 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 50100	. . . . . . . . . 10 ⊢ Ran Fn ((V × V) × V)
15	14	fndmi 6597	. . . . . . . . 9 ⊢ dom Ran = ((V × V) × V)
16	13, 15	eleqtrdi 2847	. . . . . . . 8 ⊢ (( Ran ‘⟨𝑃, 𝐸⟩) ≠ ∅ → ⟨𝑃, 𝐸⟩ ∈ ((V × V) × V))
17		opelxp1 5667	. . . . . . . 8 ⊢ (⟨𝑃, 𝐸⟩ ∈ ((V × V) × V) → 𝑃 ∈ (V × V))
18		1st2nd2 7975	. . . . . . . 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 7376	. . . . . 6 ⊢ (( Ran ‘⟨𝑃, 𝐸⟩) ≠ ∅ → (𝑃 Ran 𝐸) = (⟨(1st ‘𝑃), (2nd ‘𝑃)⟩ Ran 𝐸))
21		eqid 2737	. . . . . . 7 ⊢ ((2nd ‘𝑃) FuncCat 𝐸) = ((2nd ‘𝑃) FuncCat 𝐸)
22		eqid 2737	. . . . . . 7 ⊢ ((1st ‘𝑃) FuncCat 𝐸) = ((1st ‘𝑃) FuncCat 𝐸)
23		fvexd 6850	. . . . . . 7 ⊢ (( Ran ‘⟨𝑃, 𝐸⟩) ≠ ∅ → (1st ‘𝑃) ∈ V)
24		fvexd 6850	. . . . . . 7 ⊢ (( Ran ‘⟨𝑃, 𝐸⟩) ≠ ∅ → (2nd ‘𝑃) ∈ V)
25		opelxp2 5668	. . . . . . . 8 ⊢ (⟨𝑃, 𝐸⟩ ∈ ((V × V) × V) → 𝐸 ∈ V)
26	16, 25	syl 17	. . . . . . 7 ⊢ (( Ran ‘⟨𝑃, 𝐸⟩) ≠ ∅ → 𝐸 ∈ V)
27		eqid 2737	. . . . . . 7 ⊢ (oppCat‘((2nd ‘𝑃) FuncCat 𝐸)) = (oppCat‘((2nd ‘𝑃) FuncCat 𝐸))
28		eqid 2737	. . . . . . 7 ⊢ (oppCat‘((1st ‘𝑃) FuncCat 𝐸)) = (oppCat‘((1st ‘𝑃) FuncCat 𝐸))
29	21, 22, 23, 24, 26, 27, 28	ranfval 50104	. . . . . 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 2772	. . . . 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 5855	. . . 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 5729	. . 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 3016	1 ⊢ Rel dom (𝑃 Ran 𝐸)

Syntax hints:   = wceq 1542   ∈ wcel 2114   ≠ wne 2933  Vcvv 3430  ∅c0 4274  {csn 4568  ⟨cop 4574   × cxp 5623  dom cdm 5625   ↾ cres 5627  Rel wrel 5630  Fun wfun 6487  ‘cfv 6493  (class class class)co 7361   ∈ cmpo 7363  1^st c1st 7934  2^nd c2nd 7935  oppCatcoppc 17671   Func cfunc 17815   FuncCat cfuc 17906   oppFunc coppf 49612   UP cup 49663   −∘_F cprcof 49863   Ran cran 50096

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 5213  ax-sep 5232  ax-nul 5242  ax-pow 5303  ax-pr 5371  ax-un 7683

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 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-iun 4936  df-br 5087  df-opab 5149  df-mpt 5168  df-id 5520  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-iota 6449  df-fun 6495  df-fn 6496  df-f 6497  df-f1 6498  df-fo 6499  df-f1o 6500  df-fv 6501  df-ov 7364  df-oprab 7365  df-mpo 7366  df-1st 7936  df-2nd 7937  df-ran 50098

This theorem is referenced by: (None)