Users' Mathboxes Mathbox for Thierry Arnoux < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  prsrn Structured version   Visualization version   GIF version

Theorem prsrn 32895
Description: Range of the relation of a proset. (Contributed by Thierry Arnoux, 11-Sep-2018.)
Hypotheses
Ref Expression
ordtNEW.b 𝐡 = (Baseβ€˜πΎ)
ordtNEW.l ≀ = ((leβ€˜πΎ) ∩ (𝐡 Γ— 𝐡))
Assertion
Ref Expression
prsrn (𝐾 ∈ Proset β†’ ran ≀ = 𝐡)

Proof of Theorem prsrn
Dummy variables π‘₯ 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ordtNEW.l . . . . 5 ≀ = ((leβ€˜πΎ) ∩ (𝐡 Γ— 𝐡))
21rneqi 5937 . . . 4 ran ≀ = ran ((leβ€˜πΎ) ∩ (𝐡 Γ— 𝐡))
32eleq2i 2826 . . 3 (π‘₯ ∈ ran ≀ ↔ π‘₯ ∈ ran ((leβ€˜πΎ) ∩ (𝐡 Γ— 𝐡)))
4 vex 3479 . . . . 5 π‘₯ ∈ V
54elrn2 5893 . . . 4 (π‘₯ ∈ ran ((leβ€˜πΎ) ∩ (𝐡 Γ— 𝐡)) ↔ βˆƒπ‘¦βŸ¨π‘¦, π‘₯⟩ ∈ ((leβ€˜πΎ) ∩ (𝐡 Γ— 𝐡)))
6 ordtNEW.b . . . . . . . . . 10 𝐡 = (Baseβ€˜πΎ)
7 eqid 2733 . . . . . . . . . 10 (leβ€˜πΎ) = (leβ€˜πΎ)
86, 7prsref 18252 . . . . . . . . 9 ((𝐾 ∈ Proset ∧ π‘₯ ∈ 𝐡) β†’ π‘₯(leβ€˜πΎ)π‘₯)
9 df-br 5150 . . . . . . . . 9 (π‘₯(leβ€˜πΎ)π‘₯ ↔ ⟨π‘₯, π‘₯⟩ ∈ (leβ€˜πΎ))
108, 9sylib 217 . . . . . . . 8 ((𝐾 ∈ Proset ∧ π‘₯ ∈ 𝐡) β†’ ⟨π‘₯, π‘₯⟩ ∈ (leβ€˜πΎ))
11 simpr 486 . . . . . . . . 9 ((𝐾 ∈ Proset ∧ π‘₯ ∈ 𝐡) β†’ π‘₯ ∈ 𝐡)
1211, 11opelxpd 5716 . . . . . . . 8 ((𝐾 ∈ Proset ∧ π‘₯ ∈ 𝐡) β†’ ⟨π‘₯, π‘₯⟩ ∈ (𝐡 Γ— 𝐡))
1310, 12elind 4195 . . . . . . 7 ((𝐾 ∈ Proset ∧ π‘₯ ∈ 𝐡) β†’ ⟨π‘₯, π‘₯⟩ ∈ ((leβ€˜πΎ) ∩ (𝐡 Γ— 𝐡)))
14 opeq1 4874 . . . . . . . . 9 (𝑦 = π‘₯ β†’ βŸ¨π‘¦, π‘₯⟩ = ⟨π‘₯, π‘₯⟩)
1514eleq1d 2819 . . . . . . . 8 (𝑦 = π‘₯ β†’ (βŸ¨π‘¦, π‘₯⟩ ∈ ((leβ€˜πΎ) ∩ (𝐡 Γ— 𝐡)) ↔ ⟨π‘₯, π‘₯⟩ ∈ ((leβ€˜πΎ) ∩ (𝐡 Γ— 𝐡))))
164, 15spcev 3597 . . . . . . 7 (⟨π‘₯, π‘₯⟩ ∈ ((leβ€˜πΎ) ∩ (𝐡 Γ— 𝐡)) β†’ βˆƒπ‘¦βŸ¨π‘¦, π‘₯⟩ ∈ ((leβ€˜πΎ) ∩ (𝐡 Γ— 𝐡)))
1713, 16syl 17 . . . . . 6 ((𝐾 ∈ Proset ∧ π‘₯ ∈ 𝐡) β†’ βˆƒπ‘¦βŸ¨π‘¦, π‘₯⟩ ∈ ((leβ€˜πΎ) ∩ (𝐡 Γ— 𝐡)))
1817ex 414 . . . . 5 (𝐾 ∈ Proset β†’ (π‘₯ ∈ 𝐡 β†’ βˆƒπ‘¦βŸ¨π‘¦, π‘₯⟩ ∈ ((leβ€˜πΎ) ∩ (𝐡 Γ— 𝐡))))
19 elinel2 4197 . . . . . . 7 (βŸ¨π‘¦, π‘₯⟩ ∈ ((leβ€˜πΎ) ∩ (𝐡 Γ— 𝐡)) β†’ βŸ¨π‘¦, π‘₯⟩ ∈ (𝐡 Γ— 𝐡))
20 opelxp2 5720 . . . . . . 7 (βŸ¨π‘¦, π‘₯⟩ ∈ (𝐡 Γ— 𝐡) β†’ π‘₯ ∈ 𝐡)
2119, 20syl 17 . . . . . 6 (βŸ¨π‘¦, π‘₯⟩ ∈ ((leβ€˜πΎ) ∩ (𝐡 Γ— 𝐡)) β†’ π‘₯ ∈ 𝐡)
2221exlimiv 1934 . . . . 5 (βˆƒπ‘¦βŸ¨π‘¦, π‘₯⟩ ∈ ((leβ€˜πΎ) ∩ (𝐡 Γ— 𝐡)) β†’ π‘₯ ∈ 𝐡)
2318, 22impbid1 224 . . . 4 (𝐾 ∈ Proset β†’ (π‘₯ ∈ 𝐡 ↔ βˆƒπ‘¦βŸ¨π‘¦, π‘₯⟩ ∈ ((leβ€˜πΎ) ∩ (𝐡 Γ— 𝐡))))
245, 23bitr4id 290 . . 3 (𝐾 ∈ Proset β†’ (π‘₯ ∈ ran ((leβ€˜πΎ) ∩ (𝐡 Γ— 𝐡)) ↔ π‘₯ ∈ 𝐡))
253, 24bitrid 283 . 2 (𝐾 ∈ Proset β†’ (π‘₯ ∈ ran ≀ ↔ π‘₯ ∈ 𝐡))
2625eqrdv 2731 1 (𝐾 ∈ Proset β†’ ran ≀ = 𝐡)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 397   = wceq 1542  βˆƒwex 1782   ∈ wcel 2107   ∩ cin 3948  βŸ¨cop 4635   class class class wbr 5149   Γ— cxp 5675  ran crn 5678  β€˜cfv 6544  Basecbs 17144  lecple 17204   Proset cproset 18246
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pr 5428
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-sbc 3779  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-opab 5212  df-xp 5683  df-cnv 5685  df-dm 5687  df-rn 5688  df-iota 6496  df-fv 6552  df-proset 18248
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator