Theorem prsrn 32884

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 5935	. . . 4 ⊢ ran ≤ = ran ((le‘𝐾) ∩ (𝐵 × 𝐵))
3	2	eleq2i 2826	. . 3 ⊢ (𝑥 ∈ ran ≤ ↔ 𝑥 ∈ ran ((le‘𝐾) ∩ (𝐵 × 𝐵)))
4		vex 3479	. . . . 5 ⊢ 𝑥 ∈ V
5	4	elrn2 5891	. . . 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 18249	. . . . . . . . 9 ⊢ ((𝐾 ∈ Proset ∧ 𝑥 ∈ 𝐵) → 𝑥(le‘𝐾)𝑥)
9		df-br 5149	. . . . . . . . 9 ⊢ (𝑥(le‘𝐾)𝑥 ↔ ⟨𝑥, 𝑥⟩ ∈ (le‘𝐾))
10	8, 9	sylib 217	. . . . . . . 8 ⊢ ((𝐾 ∈ Proset ∧ 𝑥 ∈ 𝐵) → ⟨𝑥, 𝑥⟩ ∈ (le‘𝐾))
11		simpr 486	. . . . . . . . 9 ⊢ ((𝐾 ∈ Proset ∧ 𝑥 ∈ 𝐵) → 𝑥 ∈ 𝐵)
12	11, 11	opelxpd 5714	. . . . . . . 8 ⊢ ((𝐾 ∈ Proset ∧ 𝑥 ∈ 𝐵) → ⟨𝑥, 𝑥⟩ ∈ (𝐵 × 𝐵))
13	10, 12	elind 4194	. . . . . . 7 ⊢ ((𝐾 ∈ Proset ∧ 𝑥 ∈ 𝐵) → ⟨𝑥, 𝑥⟩ ∈ ((le‘𝐾) ∩ (𝐵 × 𝐵)))
14		opeq1 4873	. . . . . . . . 9 ⊢ (𝑦 = 𝑥 → ⟨𝑦, 𝑥⟩ = ⟨𝑥, 𝑥⟩)
15	14	eleq1d 2819	. . . . . . . 8 ⊢ (𝑦 = 𝑥 → (⟨𝑦, 𝑥⟩ ∈ ((le‘𝐾) ∩ (𝐵 × 𝐵)) ↔ ⟨𝑥, 𝑥⟩ ∈ ((le‘𝐾) ∩ (𝐵 × 𝐵))))
16	4, 15	spcev 3597	. . . . . . 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 414	. . . . 5 ⊢ (𝐾 ∈ Proset → (𝑥 ∈ 𝐵 → ∃𝑦⟨𝑦, 𝑥⟩ ∈ ((le‘𝐾) ∩ (𝐵 × 𝐵))))
19		elinel2 4196	. . . . . . 7 ⊢ (⟨𝑦, 𝑥⟩ ∈ ((le‘𝐾) ∩ (𝐵 × 𝐵)) → ⟨𝑦, 𝑥⟩ ∈ (𝐵 × 𝐵))
20		opelxp2 5718	. . . . . . 7 ⊢ (⟨𝑦, 𝑥⟩ ∈ (𝐵 × 𝐵) → 𝑥 ∈ 𝐵)
21	19, 20	syl 17	. . . . . 6 ⊢ (⟨𝑦, 𝑥⟩ ∈ ((le‘𝐾) ∩ (𝐵 × 𝐵)) → 𝑥 ∈ 𝐵)
22	21	exlimiv 1934	. . . . 5 ⊢ (∃𝑦⟨𝑦, 𝑥⟩ ∈ ((le‘𝐾) ∩ (𝐵 × 𝐵)) → 𝑥 ∈ 𝐵)
23	18, 22	impbid1 224	. . . 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 397   = wceq 1542  ∃wex 1782   ∈ wcel 2107   ∩ cin 3947  ⟨cop 4634   class class class wbr 5148   × cxp 5674  ran crn 5677  ‘cfv 6541  Basecbs 17141  lecple 17201   Proset cproset 18243

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704  ax-sep 5299  ax-nul 5306  ax-pr 5427

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-sbc 3778  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-xp 5682  df-cnv 5684  df-dm 5686  df-rn 5687  df-iota 6493  df-fv 6549  df-proset 18245

This theorem is referenced by: (None)