Theorem prsrn 30775

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 5689	. . . 4 ⊢ ran ≤ = ran ((le‘𝐾) ∩ (𝐵 × 𝐵))
3	2	eleq2i 2874	. . 3 ⊢ (𝑥 ∈ ran ≤ ↔ 𝑥 ∈ ran ((le‘𝐾) ∩ (𝐵 × 𝐵)))
4		ordtNEW.b	. . . . . . . . . 10 ⊢ 𝐵 = (Base‘𝐾)
5		eqid 2795	. . . . . . . . . 10 ⊢ (le‘𝐾) = (le‘𝐾)
6	4, 5	prsref 17371	. . . . . . . . 9 ⊢ ((𝐾 ∈ Proset ∧ 𝑥 ∈ 𝐵) → 𝑥(le‘𝐾)𝑥)
7		df-br 4963	. . . . . . . . 9 ⊢ (𝑥(le‘𝐾)𝑥 ↔ ⟨𝑥, 𝑥⟩ ∈ (le‘𝐾))
8	6, 7	sylib 219	. . . . . . . 8 ⊢ ((𝐾 ∈ Proset ∧ 𝑥 ∈ 𝐵) → ⟨𝑥, 𝑥⟩ ∈ (le‘𝐾))
9		simpr 485	. . . . . . . . 9 ⊢ ((𝐾 ∈ Proset ∧ 𝑥 ∈ 𝐵) → 𝑥 ∈ 𝐵)
10	9, 9	opelxpd 5481	. . . . . . . 8 ⊢ ((𝐾 ∈ Proset ∧ 𝑥 ∈ 𝐵) → ⟨𝑥, 𝑥⟩ ∈ (𝐵 × 𝐵))
11	8, 10	elind 4092	. . . . . . 7 ⊢ ((𝐾 ∈ Proset ∧ 𝑥 ∈ 𝐵) → ⟨𝑥, 𝑥⟩ ∈ ((le‘𝐾) ∩ (𝐵 × 𝐵)))
12		vex 3440	. . . . . . . 8 ⊢ 𝑥 ∈ V
13		opeq1 4710	. . . . . . . . 9 ⊢ (𝑦 = 𝑥 → ⟨𝑦, 𝑥⟩ = ⟨𝑥, 𝑥⟩)
14	13	eleq1d 2867	. . . . . . . 8 ⊢ (𝑦 = 𝑥 → (⟨𝑦, 𝑥⟩ ∈ ((le‘𝐾) ∩ (𝐵 × 𝐵)) ↔ ⟨𝑥, 𝑥⟩ ∈ ((le‘𝐾) ∩ (𝐵 × 𝐵))))
15	12, 14	spcev 3549	. . . . . . 7 ⊢ (⟨𝑥, 𝑥⟩ ∈ ((le‘𝐾) ∩ (𝐵 × 𝐵)) → ∃𝑦⟨𝑦, 𝑥⟩ ∈ ((le‘𝐾) ∩ (𝐵 × 𝐵)))
16	11, 15	syl 17	. . . . . 6 ⊢ ((𝐾 ∈ Proset ∧ 𝑥 ∈ 𝐵) → ∃𝑦⟨𝑦, 𝑥⟩ ∈ ((le‘𝐾) ∩ (𝐵 × 𝐵)))
17	16	ex 413	. . . . 5 ⊢ (𝐾 ∈ Proset → (𝑥 ∈ 𝐵 → ∃𝑦⟨𝑦, 𝑥⟩ ∈ ((le‘𝐾) ∩ (𝐵 × 𝐵))))
18		elinel2 4094	. . . . . . 7 ⊢ (⟨𝑦, 𝑥⟩ ∈ ((le‘𝐾) ∩ (𝐵 × 𝐵)) → ⟨𝑦, 𝑥⟩ ∈ (𝐵 × 𝐵))
19		opelxp2 5485	. . . . . . 7 ⊢ (⟨𝑦, 𝑥⟩ ∈ (𝐵 × 𝐵) → 𝑥 ∈ 𝐵)
20	18, 19	syl 17	. . . . . 6 ⊢ (⟨𝑦, 𝑥⟩ ∈ ((le‘𝐾) ∩ (𝐵 × 𝐵)) → 𝑥 ∈ 𝐵)
21	20	exlimiv 1908	. . . . 5 ⊢ (∃𝑦⟨𝑦, 𝑥⟩ ∈ ((le‘𝐾) ∩ (𝐵 × 𝐵)) → 𝑥 ∈ 𝐵)
22	17, 21	impbid1 226	. . . 4 ⊢ (𝐾 ∈ Proset → (𝑥 ∈ 𝐵 ↔ ∃𝑦⟨𝑦, 𝑥⟩ ∈ ((le‘𝐾) ∩ (𝐵 × 𝐵))))
23	12	elrn2 5703	. . . 4 ⊢ (𝑥 ∈ ran ((le‘𝐾) ∩ (𝐵 × 𝐵)) ↔ ∃𝑦⟨𝑦, 𝑥⟩ ∈ ((le‘𝐾) ∩ (𝐵 × 𝐵)))
24	22, 23	syl6rbbr 291	. . 3 ⊢ (𝐾 ∈ Proset → (𝑥 ∈ ran ((le‘𝐾) ∩ (𝐵 × 𝐵)) ↔ 𝑥 ∈ 𝐵))
25	3, 24	syl5bb 284	. 2 ⊢ (𝐾 ∈ Proset → (𝑥 ∈ ran ≤ ↔ 𝑥 ∈ 𝐵))
26	25	eqrdv 2793	1 ⊢ (𝐾 ∈ Proset → ran ≤ = 𝐵)

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1522  ∃wex 1761   ∈ wcel 2081   ∩ cin 3858  ⟨cop 4478   class class class wbr 4962   × cxp 5441  ran crn 5444  ‘cfv 6225  Basecbs 16312  lecple 16401   Proset cproset 17365

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1777  ax-4 1791  ax-5 1888  ax-6 1947  ax-7 1992  ax-8 2083  ax-9 2091  ax-10 2112  ax-11 2126  ax-12 2141  ax-13 2344  ax-ext 2769  ax-sep 5094  ax-nul 5101  ax-pr 5221

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3an 1082  df-tru 1525  df-ex 1762  df-nf 1766  df-sb 2043  df-mo 2576  df-eu 2612  df-clab 2776  df-cleq 2788  df-clel 2863  df-nfc 2935  df-ral 3110  df-rex 3111  df-rab 3114  df-v 3439  df-sbc 3707  df-dif 3862  df-un 3864  df-in 3866  df-ss 3874  df-nul 4212  df-if 4382  df-sn 4473  df-pr 4475  df-op 4479  df-uni 4746  df-br 4963  df-opab 5025  df-xp 5449  df-cnv 5451  df-dm 5453  df-rn 5454  df-iota 6189  df-fv 6233  df-proset 17367

This theorem is referenced by: (None)