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Theorem prsrn 31266
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 5775 . . . 4 ran = ran ((le‘𝐾) ∩ (𝐵 × 𝐵))
32eleq2i 2884 . . 3 (𝑥 ∈ ran 𝑥 ∈ ran ((le‘𝐾) ∩ (𝐵 × 𝐵)))
4 ordtNEW.b . . . . . . . . . 10 𝐵 = (Base‘𝐾)
5 eqid 2801 . . . . . . . . . 10 (le‘𝐾) = (le‘𝐾)
64, 5prsref 17537 . . . . . . . . 9 ((𝐾 ∈ Proset ∧ 𝑥𝐵) → 𝑥(le‘𝐾)𝑥)
7 df-br 5034 . . . . . . . . 9 (𝑥(le‘𝐾)𝑥 ↔ ⟨𝑥, 𝑥⟩ ∈ (le‘𝐾))
86, 7sylib 221 . . . . . . . 8 ((𝐾 ∈ Proset ∧ 𝑥𝐵) → ⟨𝑥, 𝑥⟩ ∈ (le‘𝐾))
9 simpr 488 . . . . . . . . 9 ((𝐾 ∈ Proset ∧ 𝑥𝐵) → 𝑥𝐵)
109, 9opelxpd 5561 . . . . . . . 8 ((𝐾 ∈ Proset ∧ 𝑥𝐵) → ⟨𝑥, 𝑥⟩ ∈ (𝐵 × 𝐵))
118, 10elind 4124 . . . . . . 7 ((𝐾 ∈ Proset ∧ 𝑥𝐵) → ⟨𝑥, 𝑥⟩ ∈ ((le‘𝐾) ∩ (𝐵 × 𝐵)))
12 vex 3447 . . . . . . . 8 𝑥 ∈ V
13 opeq1 4766 . . . . . . . . 9 (𝑦 = 𝑥 → ⟨𝑦, 𝑥⟩ = ⟨𝑥, 𝑥⟩)
1413eleq1d 2877 . . . . . . . 8 (𝑦 = 𝑥 → (⟨𝑦, 𝑥⟩ ∈ ((le‘𝐾) ∩ (𝐵 × 𝐵)) ↔ ⟨𝑥, 𝑥⟩ ∈ ((le‘𝐾) ∩ (𝐵 × 𝐵))))
1512, 14spcev 3558 . . . . . . 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 416 . . . . 5 (𝐾 ∈ Proset → (𝑥𝐵 → ∃𝑦𝑦, 𝑥⟩ ∈ ((le‘𝐾) ∩ (𝐵 × 𝐵))))
18 elinel2 4126 . . . . . . 7 (⟨𝑦, 𝑥⟩ ∈ ((le‘𝐾) ∩ (𝐵 × 𝐵)) → ⟨𝑦, 𝑥⟩ ∈ (𝐵 × 𝐵))
19 opelxp2 5565 . . . . . . 7 (⟨𝑦, 𝑥⟩ ∈ (𝐵 × 𝐵) → 𝑥𝐵)
2018, 19syl 17 . . . . . 6 (⟨𝑦, 𝑥⟩ ∈ ((le‘𝐾) ∩ (𝐵 × 𝐵)) → 𝑥𝐵)
2120exlimiv 1931 . . . . 5 (∃𝑦𝑦, 𝑥⟩ ∈ ((le‘𝐾) ∩ (𝐵 × 𝐵)) → 𝑥𝐵)
2217, 21impbid1 228 . . . 4 (𝐾 ∈ Proset → (𝑥𝐵 ↔ ∃𝑦𝑦, 𝑥⟩ ∈ ((le‘𝐾) ∩ (𝐵 × 𝐵))))
2312elrn2 5789 . . . 4 (𝑥 ∈ ran ((le‘𝐾) ∩ (𝐵 × 𝐵)) ↔ ∃𝑦𝑦, 𝑥⟩ ∈ ((le‘𝐾) ∩ (𝐵 × 𝐵)))
2422, 23syl6rbbr 293 . . 3 (𝐾 ∈ Proset → (𝑥 ∈ ran ((le‘𝐾) ∩ (𝐵 × 𝐵)) ↔ 𝑥𝐵))
253, 24syl5bb 286 . 2 (𝐾 ∈ Proset → (𝑥 ∈ ran 𝑥𝐵))
2625eqrdv 2799 1 (𝐾 ∈ Proset → ran = 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399   = wceq 1538  wex 1781  wcel 2112  cin 3883  cop 4534   class class class wbr 5033   × cxp 5521  ran crn 5524  cfv 6328  Basecbs 16478  lecple 16567   Proset cproset 17531
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 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-sep 5170  ax-nul 5177  ax-pr 5298
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ral 3114  df-rex 3115  df-v 3446  df-sbc 3724  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4804  df-br 5034  df-opab 5096  df-xp 5529  df-cnv 5531  df-dm 5533  df-rn 5534  df-iota 6287  df-fv 6336  df-proset 17533
This theorem is referenced by: (None)
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