Theorem prsrn 33861

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 5962	. . . 4 ⊢ ran ≤ = ran ((le‘𝐾) ∩ (𝐵 × 𝐵))
3	2	eleq2i 2836	. . 3 ⊢ (𝑥 ∈ ran ≤ ↔ 𝑥 ∈ ran ((le‘𝐾) ∩ (𝐵 × 𝐵)))
4		vex 3492	. . . . 5 ⊢ 𝑥 ∈ V
5	4	elrn2 5917	. . . 4 ⊢ (𝑥 ∈ ran ((le‘𝐾) ∩ (𝐵 × 𝐵)) ↔ ∃𝑦⟨𝑦, 𝑥⟩ ∈ ((le‘𝐾) ∩ (𝐵 × 𝐵)))
6		ordtNEW.b	. . . . . . . . . 10 ⊢ 𝐵 = (Base‘𝐾)
7		eqid 2740	. . . . . . . . . 10 ⊢ (le‘𝐾) = (le‘𝐾)
8	6, 7	prsref 18369	. . . . . . . . 9 ⊢ ((𝐾 ∈ Proset ∧ 𝑥 ∈ 𝐵) → 𝑥(le‘𝐾)𝑥)
9		df-br 5167	. . . . . . . . 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 5739	. . . . . . . 8 ⊢ ((𝐾 ∈ Proset ∧ 𝑥 ∈ 𝐵) → ⟨𝑥, 𝑥⟩ ∈ (𝐵 × 𝐵))
13	10, 12	elind 4223	. . . . . . 7 ⊢ ((𝐾 ∈ Proset ∧ 𝑥 ∈ 𝐵) → ⟨𝑥, 𝑥⟩ ∈ ((le‘𝐾) ∩ (𝐵 × 𝐵)))
14		opeq1 4897	. . . . . . . . 9 ⊢ (𝑦 = 𝑥 → ⟨𝑦, 𝑥⟩ = ⟨𝑥, 𝑥⟩)
15	14	eleq1d 2829	. . . . . . . 8 ⊢ (𝑦 = 𝑥 → (⟨𝑦, 𝑥⟩ ∈ ((le‘𝐾) ∩ (𝐵 × 𝐵)) ↔ ⟨𝑥, 𝑥⟩ ∈ ((le‘𝐾) ∩ (𝐵 × 𝐵))))
16	4, 15	spcev 3619	. . . . . . 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 4225	. . . . . . 7 ⊢ (⟨𝑦, 𝑥⟩ ∈ ((le‘𝐾) ∩ (𝐵 × 𝐵)) → ⟨𝑦, 𝑥⟩ ∈ (𝐵 × 𝐵))
20		opelxp2 5743	. . . . . . 7 ⊢ (⟨𝑦, 𝑥⟩ ∈ (𝐵 × 𝐵) → 𝑥 ∈ 𝐵)
21	19, 20	syl 17	. . . . . 6 ⊢ (⟨𝑦, 𝑥⟩ ∈ ((le‘𝐾) ∩ (𝐵 × 𝐵)) → 𝑥 ∈ 𝐵)
22	21	exlimiv 1929	. . . . 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 2738	1 ⊢ (𝐾 ∈ Proset → ran ≤ = 𝐵)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1537  ∃wex 1777   ∈ wcel 2108   ∩ cin 3975  ⟨cop 4654   class class class wbr 5166   × cxp 5698  ran crn 5701  ‘cfv 6573  Basecbs 17258  lecple 17318   Proset cproset 18363

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-ne 2947  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-sbc 3805  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-opab 5229  df-xp 5706  df-cnv 5708  df-dm 5710  df-rn 5711  df-iota 6525  df-fv 6581  df-proset 18365

This theorem is referenced by: (None)