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Theorem prsrn 32227
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 5885 . . . 4 ran = ran ((le‘𝐾) ∩ (𝐵 × 𝐵))
32eleq2i 2829 . . 3 (𝑥 ∈ ran 𝑥 ∈ ran ((le‘𝐾) ∩ (𝐵 × 𝐵)))
4 vex 3447 . . . . 5 𝑥 ∈ V
54elrn2 5841 . . . 4 (𝑥 ∈ ran ((le‘𝐾) ∩ (𝐵 × 𝐵)) ↔ ∃𝑦𝑦, 𝑥⟩ ∈ ((le‘𝐾) ∩ (𝐵 × 𝐵)))
6 ordtNEW.b . . . . . . . . . 10 𝐵 = (Base‘𝐾)
7 eqid 2737 . . . . . . . . . 10 (le‘𝐾) = (le‘𝐾)
86, 7prsref 18119 . . . . . . . . 9 ((𝐾 ∈ Proset ∧ 𝑥𝐵) → 𝑥(le‘𝐾)𝑥)
9 df-br 5101 . . . . . . . . 9 (𝑥(le‘𝐾)𝑥 ↔ ⟨𝑥, 𝑥⟩ ∈ (le‘𝐾))
108, 9sylib 217 . . . . . . . 8 ((𝐾 ∈ Proset ∧ 𝑥𝐵) → ⟨𝑥, 𝑥⟩ ∈ (le‘𝐾))
11 simpr 486 . . . . . . . . 9 ((𝐾 ∈ Proset ∧ 𝑥𝐵) → 𝑥𝐵)
1211, 11opelxpd 5665 . . . . . . . 8 ((𝐾 ∈ Proset ∧ 𝑥𝐵) → ⟨𝑥, 𝑥⟩ ∈ (𝐵 × 𝐵))
1310, 12elind 4149 . . . . . . 7 ((𝐾 ∈ Proset ∧ 𝑥𝐵) → ⟨𝑥, 𝑥⟩ ∈ ((le‘𝐾) ∩ (𝐵 × 𝐵)))
14 opeq1 4825 . . . . . . . . 9 (𝑦 = 𝑥 → ⟨𝑦, 𝑥⟩ = ⟨𝑥, 𝑥⟩)
1514eleq1d 2822 . . . . . . . 8 (𝑦 = 𝑥 → (⟨𝑦, 𝑥⟩ ∈ ((le‘𝐾) ∩ (𝐵 × 𝐵)) ↔ ⟨𝑥, 𝑥⟩ ∈ ((le‘𝐾) ∩ (𝐵 × 𝐵))))
164, 15spcev 3560 . . . . . . 7 (⟨𝑥, 𝑥⟩ ∈ ((le‘𝐾) ∩ (𝐵 × 𝐵)) → ∃𝑦𝑦, 𝑥⟩ ∈ ((le‘𝐾) ∩ (𝐵 × 𝐵)))
1713, 16syl 17 . . . . . 6 ((𝐾 ∈ Proset ∧ 𝑥𝐵) → ∃𝑦𝑦, 𝑥⟩ ∈ ((le‘𝐾) ∩ (𝐵 × 𝐵)))
1817ex 414 . . . . 5 (𝐾 ∈ Proset → (𝑥𝐵 → ∃𝑦𝑦, 𝑥⟩ ∈ ((le‘𝐾) ∩ (𝐵 × 𝐵))))
19 elinel2 4151 . . . . . . 7 (⟨𝑦, 𝑥⟩ ∈ ((le‘𝐾) ∩ (𝐵 × 𝐵)) → ⟨𝑦, 𝑥⟩ ∈ (𝐵 × 𝐵))
20 opelxp2 5669 . . . . . . 7 (⟨𝑦, 𝑥⟩ ∈ (𝐵 × 𝐵) → 𝑥𝐵)
2119, 20syl 17 . . . . . 6 (⟨𝑦, 𝑥⟩ ∈ ((le‘𝐾) ∩ (𝐵 × 𝐵)) → 𝑥𝐵)
2221exlimiv 1933 . . . . 5 (∃𝑦𝑦, 𝑥⟩ ∈ ((le‘𝐾) ∩ (𝐵 × 𝐵)) → 𝑥𝐵)
2318, 22impbid1 224 . . . 4 (𝐾 ∈ Proset → (𝑥𝐵 ↔ ∃𝑦𝑦, 𝑥⟩ ∈ ((le‘𝐾) ∩ (𝐵 × 𝐵))))
245, 23bitr4id 290 . . 3 (𝐾 ∈ Proset → (𝑥 ∈ ran ((le‘𝐾) ∩ (𝐵 × 𝐵)) ↔ 𝑥𝐵))
253, 24bitrid 283 . 2 (𝐾 ∈ Proset → (𝑥 ∈ ran 𝑥𝐵))
2625eqrdv 2735 1 (𝐾 ∈ Proset → ran = 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397   = wceq 1541  wex 1781  wcel 2106  cin 3904  cop 4587   class class class wbr 5100   × cxp 5625  ran crn 5628  cfv 6488  Basecbs 17014  lecple 17071   Proset cproset 18113
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2708  ax-sep 5251  ax-nul 5258  ax-pr 5379
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2715  df-cleq 2729  df-clel 2815  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3406  df-v 3445  df-sbc 3735  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4278  df-if 4482  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4861  df-br 5101  df-opab 5163  df-xp 5633  df-cnv 5635  df-dm 5637  df-rn 5638  df-iota 6440  df-fv 6496  df-proset 18115
This theorem is referenced by: (None)
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