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Theorem prsrn 29743
Description: Range of the relation of a preset. (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 ∈ Preset → 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 5312 . . . 4 ran = ran ((le‘𝐾) ∩ (𝐵 × 𝐵))
32eleq2i 2690 . . 3 (𝑥 ∈ ran 𝑥 ∈ ran ((le‘𝐾) ∩ (𝐵 × 𝐵)))
4 ordtNEW.b . . . . . . . . . 10 𝐵 = (Base‘𝐾)
5 eqid 2621 . . . . . . . . . 10 (le‘𝐾) = (le‘𝐾)
64, 5prsref 16853 . . . . . . . . 9 ((𝐾 ∈ Preset ∧ 𝑥𝐵) → 𝑥(le‘𝐾)𝑥)
7 df-br 4614 . . . . . . . . 9 (𝑥(le‘𝐾)𝑥 ↔ ⟨𝑥, 𝑥⟩ ∈ (le‘𝐾))
86, 7sylib 208 . . . . . . . 8 ((𝐾 ∈ Preset ∧ 𝑥𝐵) → ⟨𝑥, 𝑥⟩ ∈ (le‘𝐾))
9 simpr 477 . . . . . . . . 9 ((𝐾 ∈ Preset ∧ 𝑥𝐵) → 𝑥𝐵)
10 opelxpi 5108 . . . . . . . . 9 ((𝑥𝐵𝑥𝐵) → ⟨𝑥, 𝑥⟩ ∈ (𝐵 × 𝐵))
119, 10sylancom 700 . . . . . . . 8 ((𝐾 ∈ Preset ∧ 𝑥𝐵) → ⟨𝑥, 𝑥⟩ ∈ (𝐵 × 𝐵))
128, 11elind 3776 . . . . . . 7 ((𝐾 ∈ Preset ∧ 𝑥𝐵) → ⟨𝑥, 𝑥⟩ ∈ ((le‘𝐾) ∩ (𝐵 × 𝐵)))
13 vex 3189 . . . . . . . 8 𝑥 ∈ V
14 opeq1 4370 . . . . . . . . 9 (𝑦 = 𝑥 → ⟨𝑦, 𝑥⟩ = ⟨𝑥, 𝑥⟩)
1514eleq1d 2683 . . . . . . . 8 (𝑦 = 𝑥 → (⟨𝑦, 𝑥⟩ ∈ ((le‘𝐾) ∩ (𝐵 × 𝐵)) ↔ ⟨𝑥, 𝑥⟩ ∈ ((le‘𝐾) ∩ (𝐵 × 𝐵))))
1613, 15spcev 3286 . . . . . . 7 (⟨𝑥, 𝑥⟩ ∈ ((le‘𝐾) ∩ (𝐵 × 𝐵)) → ∃𝑦𝑦, 𝑥⟩ ∈ ((le‘𝐾) ∩ (𝐵 × 𝐵)))
1712, 16syl 17 . . . . . 6 ((𝐾 ∈ Preset ∧ 𝑥𝐵) → ∃𝑦𝑦, 𝑥⟩ ∈ ((le‘𝐾) ∩ (𝐵 × 𝐵)))
1817ex 450 . . . . 5 (𝐾 ∈ Preset → (𝑥𝐵 → ∃𝑦𝑦, 𝑥⟩ ∈ ((le‘𝐾) ∩ (𝐵 × 𝐵))))
19 inss2 3812 . . . . . . . 8 ((le‘𝐾) ∩ (𝐵 × 𝐵)) ⊆ (𝐵 × 𝐵)
2019sseli 3579 . . . . . . 7 (⟨𝑦, 𝑥⟩ ∈ ((le‘𝐾) ∩ (𝐵 × 𝐵)) → ⟨𝑦, 𝑥⟩ ∈ (𝐵 × 𝐵))
21 opelxp2 5111 . . . . . . 7 (⟨𝑦, 𝑥⟩ ∈ (𝐵 × 𝐵) → 𝑥𝐵)
2220, 21syl 17 . . . . . 6 (⟨𝑦, 𝑥⟩ ∈ ((le‘𝐾) ∩ (𝐵 × 𝐵)) → 𝑥𝐵)
2322exlimiv 1855 . . . . 5 (∃𝑦𝑦, 𝑥⟩ ∈ ((le‘𝐾) ∩ (𝐵 × 𝐵)) → 𝑥𝐵)
2418, 23impbid1 215 . . . 4 (𝐾 ∈ Preset → (𝑥𝐵 ↔ ∃𝑦𝑦, 𝑥⟩ ∈ ((le‘𝐾) ∩ (𝐵 × 𝐵))))
2513elrn2 5325 . . . 4 (𝑥 ∈ ran ((le‘𝐾) ∩ (𝐵 × 𝐵)) ↔ ∃𝑦𝑦, 𝑥⟩ ∈ ((le‘𝐾) ∩ (𝐵 × 𝐵)))
2624, 25syl6rbbr 279 . . 3 (𝐾 ∈ Preset → (𝑥 ∈ ran ((le‘𝐾) ∩ (𝐵 × 𝐵)) ↔ 𝑥𝐵))
273, 26syl5bb 272 . 2 (𝐾 ∈ Preset → (𝑥 ∈ ran 𝑥𝐵))
2827eqrdv 2619 1 (𝐾 ∈ Preset → ran = 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1480  wex 1701  wcel 1987  cin 3554  cop 4154   class class class wbr 4613   × cxp 5072  ran crn 5075  cfv 5847  Basecbs 15781  lecple 15869   Preset cpreset 16847
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pr 4867
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-sbc 3418  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-br 4614  df-opab 4674  df-xp 5080  df-cnv 5082  df-dm 5084  df-rn 5085  df-iota 5810  df-fv 5855  df-preset 16849
This theorem is referenced by: (None)
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