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Theorem p1val 18473
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 3501 . 2 (𝐾𝑉𝐾 ∈ V)
2 p1val.t . . 3 1 = (1.‘𝐾)
3 fveq2 6906 . . . . . 6 (𝑘 = 𝐾 → (lub‘𝑘) = (lub‘𝐾))
4 p1val.u . . . . . 6 𝑈 = (lub‘𝐾)
53, 4eqtr4di 2795 . . . . 5 (𝑘 = 𝐾 → (lub‘𝑘) = 𝑈)
6 fveq2 6906 . . . . . 6 (𝑘 = 𝐾 → (Base‘𝑘) = (Base‘𝐾))
7 p1val.b . . . . . 6 𝐵 = (Base‘𝐾)
86, 7eqtr4di 2795 . . . . 5 (𝑘 = 𝐾 → (Base‘𝑘) = 𝐵)
95, 8fveq12d 6913 . . . 4 (𝑘 = 𝐾 → ((lub‘𝑘)‘(Base‘𝑘)) = (𝑈𝐵))
10 df-p1 18471 . . . 4 1. = (𝑘 ∈ V ↦ ((lub‘𝑘)‘(Base‘𝑘)))
11 fvex 6919 . . . 4 (𝑈𝐵) ∈ V
129, 10, 11fvmpt 7016 . . 3 (𝐾 ∈ V → (1.‘𝐾) = (𝑈𝐵))
132, 12eqtrid 2789 . 2 (𝐾 ∈ V → 1 = (𝑈𝐵))
141, 13syl 17 1 (𝐾𝑉1 = (𝑈𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1540  wcel 2108  Vcvv 3480  cfv 6561  Basecbs 17247  lubclub 18355  1.cp1 18469
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-iota 6514  df-fun 6563  df-fv 6569  df-p1 18471
This theorem is referenced by:  ple1  18475  clatp1cl  32967  xrsp1  33015  op1cl  39186
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