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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.
StepHypRef Expression
1 ordtNEW.l . . . . 5 = ((le‘𝐾) ∩ (𝐵 × 𝐵))
21rneqi 5689 . . . 4 ran = ran ((le‘𝐾) ∩ (𝐵 × 𝐵))
32eleq2i 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)
64, 5prsref 17371 . . . . . . . . 9 ((𝐾 ∈ Proset ∧ 𝑥𝐵) → 𝑥(le‘𝐾)𝑥)
7 df-br 4963 . . . . . . . . 9 (𝑥(le‘𝐾)𝑥 ↔ ⟨𝑥, 𝑥⟩ ∈ (le‘𝐾))
86, 7sylib 219 . . . . . . . 8 ((𝐾 ∈ Proset ∧ 𝑥𝐵) → ⟨𝑥, 𝑥⟩ ∈ (le‘𝐾))
9 simpr 485 . . . . . . . . 9 ((𝐾 ∈ Proset ∧ 𝑥𝐵) → 𝑥𝐵)
109, 9opelxpd 5481 . . . . . . . 8 ((𝐾 ∈ Proset ∧ 𝑥𝐵) → ⟨𝑥, 𝑥⟩ ∈ (𝐵 × 𝐵))
118, 10elind 4092 . . . . . . 7 ((𝐾 ∈ Proset ∧ 𝑥𝐵) → ⟨𝑥, 𝑥⟩ ∈ ((le‘𝐾) ∩ (𝐵 × 𝐵)))
12 vex 3440 . . . . . . . 8 𝑥 ∈ V
13 opeq1 4710 . . . . . . . . 9 (𝑦 = 𝑥 → ⟨𝑦, 𝑥⟩ = ⟨𝑥, 𝑥⟩)
1413eleq1d 2867 . . . . . . . 8 (𝑦 = 𝑥 → (⟨𝑦, 𝑥⟩ ∈ ((le‘𝐾) ∩ (𝐵 × 𝐵)) ↔ ⟨𝑥, 𝑥⟩ ∈ ((le‘𝐾) ∩ (𝐵 × 𝐵))))
1512, 14spcev 3549 . . . . . . 7 (⟨𝑥, 𝑥⟩ ∈ ((le‘𝐾) ∩ (𝐵 × 𝐵)) → ∃𝑦𝑦, 𝑥⟩ ∈ ((le‘𝐾) ∩ (𝐵 × 𝐵)))
1611, 15syl 17 . . . . . 6 ((𝐾 ∈ Proset ∧ 𝑥𝐵) → ∃𝑦𝑦, 𝑥⟩ ∈ ((le‘𝐾) ∩ (𝐵 × 𝐵)))
1716ex 413 . . . . 5 (𝐾 ∈ Proset → (𝑥𝐵 → ∃𝑦𝑦, 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) → 𝑥𝐵)
2018, 19syl 17 . . . . . 6 (⟨𝑦, 𝑥⟩ ∈ ((le‘𝐾) ∩ (𝐵 × 𝐵)) → 𝑥𝐵)
2120exlimiv 1908 . . . . 5 (∃𝑦𝑦, 𝑥⟩ ∈ ((le‘𝐾) ∩ (𝐵 × 𝐵)) → 𝑥𝐵)
2217, 21impbid1 226 . . . 4 (𝐾 ∈ Proset → (𝑥𝐵 ↔ ∃𝑦𝑦, 𝑥⟩ ∈ ((le‘𝐾) ∩ (𝐵 × 𝐵))))
2312elrn2 5703 . . . 4 (𝑥 ∈ ran ((le‘𝐾) ∩ (𝐵 × 𝐵)) ↔ ∃𝑦𝑦, 𝑥⟩ ∈ ((le‘𝐾) ∩ (𝐵 × 𝐵)))
2422, 23syl6rbbr 291 . . 3 (𝐾 ∈ Proset → (𝑥 ∈ ran ((le‘𝐾) ∩ (𝐵 × 𝐵)) ↔ 𝑥𝐵))
253, 24syl5bb 284 . 2 (𝐾 ∈ Proset → (𝑥 ∈ ran 𝑥𝐵))
2625eqrdv 2793 1 (𝐾 ∈ Proset → ran = 𝐵)
Colors of variables: wff setvar class
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)
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