Theorem prsrn 34000

Description: Range of the relation of a proset. (Contributed by Thierry Arnoux, 11-Sep-2018.)

Hypotheses

Ref	Expression
ordtNEW.b	⊢ 𝐵 = (Base‘𝐾)
ordtNEW.l	⊢ ≤ = ((le‘𝐾) ∩ (𝐵 × 𝐵))

Assertion

Ref	Expression
prsrn	⊢ (𝐾 ∈ Proset → ran ≤ = 𝐵)

Proof of Theorem prsrn

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

Step	Hyp	Ref	Expression
1		ordtNEW.l	. . . . 5 ⊢ ≤ = ((le‘𝐾) ∩ (𝐵 × 𝐵))
2	1	rneqi 5883	. . . 4 ⊢ ran ≤ = ran ((le‘𝐾) ∩ (𝐵 × 𝐵))
3	2	eleq2i 2825	. . 3 ⊢ (𝑥 ∈ ran ≤ ↔ 𝑥 ∈ ran ((le‘𝐾) ∩ (𝐵 × 𝐵)))
4		vex 3441	. . . . 5 ⊢ 𝑥 ∈ V
5	4	elrn2 5838	. . . 4 ⊢ (𝑥 ∈ ran ((le‘𝐾) ∩ (𝐵 × 𝐵)) ↔ ∃𝑦⟨𝑦, 𝑥⟩ ∈ ((le‘𝐾) ∩ (𝐵 × 𝐵)))
6		ordtNEW.b	. . . . . . . . . 10 ⊢ 𝐵 = (Base‘𝐾)
7		eqid 2733	. . . . . . . . . 10 ⊢ (le‘𝐾) = (le‘𝐾)
8	6, 7	prsref 18212	. . . . . . . . 9 ⊢ ((𝐾 ∈ Proset ∧ 𝑥 ∈ 𝐵) → 𝑥(le‘𝐾)𝑥)
9		df-br 5096	. . . . . . . . 9 ⊢ (𝑥(le‘𝐾)𝑥 ↔ ⟨𝑥, 𝑥⟩ ∈ (le‘𝐾))
10	8, 9	sylib 218	. . . . . . . 8 ⊢ ((𝐾 ∈ Proset ∧ 𝑥 ∈ 𝐵) → ⟨𝑥, 𝑥⟩ ∈ (le‘𝐾))
11		simpr 484	. . . . . . . . 9 ⊢ ((𝐾 ∈ Proset ∧ 𝑥 ∈ 𝐵) → 𝑥 ∈ 𝐵)
12	11, 11	opelxpd 5660	. . . . . . . 8 ⊢ ((𝐾 ∈ Proset ∧ 𝑥 ∈ 𝐵) → ⟨𝑥, 𝑥⟩ ∈ (𝐵 × 𝐵))
13	10, 12	elind 4149	. . . . . . 7 ⊢ ((𝐾 ∈ Proset ∧ 𝑥 ∈ 𝐵) → ⟨𝑥, 𝑥⟩ ∈ ((le‘𝐾) ∩ (𝐵 × 𝐵)))
14		opeq1 4826	. . . . . . . . 9 ⊢ (𝑦 = 𝑥 → ⟨𝑦, 𝑥⟩ = ⟨𝑥, 𝑥⟩)
15	14	eleq1d 2818	. . . . . . . 8 ⊢ (𝑦 = 𝑥 → (⟨𝑦, 𝑥⟩ ∈ ((le‘𝐾) ∩ (𝐵 × 𝐵)) ↔ ⟨𝑥, 𝑥⟩ ∈ ((le‘𝐾) ∩ (𝐵 × 𝐵))))
16	4, 15	spcev 3557	. . . . . . 7 ⊢ (⟨𝑥, 𝑥⟩ ∈ ((le‘𝐾) ∩ (𝐵 × 𝐵)) → ∃𝑦⟨𝑦, 𝑥⟩ ∈ ((le‘𝐾) ∩ (𝐵 × 𝐵)))
17	13, 16	syl 17	. . . . . 6 ⊢ ((𝐾 ∈ Proset ∧ 𝑥 ∈ 𝐵) → ∃𝑦⟨𝑦, 𝑥⟩ ∈ ((le‘𝐾) ∩ (𝐵 × 𝐵)))
18	17	ex 412	. . . . 5 ⊢ (𝐾 ∈ Proset → (𝑥 ∈ 𝐵 → ∃𝑦⟨𝑦, 𝑥⟩ ∈ ((le‘𝐾) ∩ (𝐵 × 𝐵))))
19		elinel2 4151	. . . . . . 7 ⊢ (⟨𝑦, 𝑥⟩ ∈ ((le‘𝐾) ∩ (𝐵 × 𝐵)) → ⟨𝑦, 𝑥⟩ ∈ (𝐵 × 𝐵))
20		opelxp2 5664	. . . . . . 7 ⊢ (⟨𝑦, 𝑥⟩ ∈ (𝐵 × 𝐵) → 𝑥 ∈ 𝐵)
21	19, 20	syl 17	. . . . . 6 ⊢ (⟨𝑦, 𝑥⟩ ∈ ((le‘𝐾) ∩ (𝐵 × 𝐵)) → 𝑥 ∈ 𝐵)
22	21	exlimiv 1931	. . . . 5 ⊢ (∃𝑦⟨𝑦, 𝑥⟩ ∈ ((le‘𝐾) ∩ (𝐵 × 𝐵)) → 𝑥 ∈ 𝐵)
23	18, 22	impbid1 225	. . . 4 ⊢ (𝐾 ∈ Proset → (𝑥 ∈ 𝐵 ↔ ∃𝑦⟨𝑦, 𝑥⟩ ∈ ((le‘𝐾) ∩ (𝐵 × 𝐵))))
24	5, 23	bitr4id 290	. . 3 ⊢ (𝐾 ∈ Proset → (𝑥 ∈ ran ((le‘𝐾) ∩ (𝐵 × 𝐵)) ↔ 𝑥 ∈ 𝐵))
25	3, 24	bitrid 283	. 2 ⊢ (𝐾 ∈ Proset → (𝑥 ∈ ran ≤ ↔ 𝑥 ∈ 𝐵))
26	25	eqrdv 2731	1 ⊢ (𝐾 ∈ Proset → ran ≤ = 𝐵)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1541  ∃wex 1780   ∈ wcel 2113   ∩ cin 3897  ⟨cop 4583   class class class wbr 5095   × cxp 5619  ran crn 5622  ‘cfv 6489  Basecbs 17127  lecple 17175   Proset cproset 18206

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-ext 2705  ax-sep 5238  ax-nul 5248  ax-pr 5374

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2712  df-cleq 2725  df-clel 2808  df-ne 2930  df-ral 3049  df-rex 3058  df-rab 3397  df-v 3439  df-sbc 3738  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4283  df-if 4477  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4861  df-br 5096  df-opab 5158  df-xp 5627  df-cnv 5629  df-dm 5631  df-rn 5632  df-iota 6445  df-fv 6497  df-proset 18208

This theorem is referenced by: (None)