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Theorem p1val 16865
 Description: Value of poset zero. (Contributed by NM, 22-Oct-2011.)
Hypotheses
Ref Expression
p1val.b 𝐵 = (Base‘𝐾)
p1val.u 𝑈 = (lub‘𝐾)
p1val.t 1 = (1.‘𝐾)
Assertion
Ref Expression
p1val (𝐾𝑉1 = (𝑈𝐵))

Proof of Theorem p1val
Dummy variable 𝑘 is distinct from all other variables.
StepHypRef Expression
1 elex 3185 . 2 (𝐾𝑉𝐾 ∈ V)
2 p1val.t . . 3 1 = (1.‘𝐾)
3 fveq2 6103 . . . . . 6 (𝑘 = 𝐾 → (lub‘𝑘) = (lub‘𝐾))
4 p1val.u . . . . . 6 𝑈 = (lub‘𝐾)
53, 4syl6eqr 2662 . . . . 5 (𝑘 = 𝐾 → (lub‘𝑘) = 𝑈)
6 fveq2 6103 . . . . . 6 (𝑘 = 𝐾 → (Base‘𝑘) = (Base‘𝐾))
7 p1val.b . . . . . 6 𝐵 = (Base‘𝐾)
86, 7syl6eqr 2662 . . . . 5 (𝑘 = 𝐾 → (Base‘𝑘) = 𝐵)
95, 8fveq12d 6109 . . . 4 (𝑘 = 𝐾 → ((lub‘𝑘)‘(Base‘𝑘)) = (𝑈𝐵))
10 df-p1 16863 . . . 4 1. = (𝑘 ∈ V ↦ ((lub‘𝑘)‘(Base‘𝑘)))
11 fvex 6113 . . . 4 (𝑈𝐵) ∈ V
129, 10, 11fvmpt 6191 . . 3 (𝐾 ∈ V → (1.‘𝐾) = (𝑈𝐵))
132, 12syl5eq 2656 . 2 (𝐾 ∈ V → 1 = (𝑈𝐵))
141, 13syl 17 1 (𝐾𝑉1 = (𝑈𝐵))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1475   ∈ wcel 1977  Vcvv 3173  ‘cfv 5804  Basecbs 15695  lubclub 16765  1.cp1 16861 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-iota 5768  df-fun 5806  df-fv 5812  df-p1 16863 This theorem is referenced by:  ple1  16867  clatp1cl  29003  xrsp1  29013  op1cl  33490
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