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Theorem p1val 18377
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 3493 . 2 (𝐾𝑉𝐾 ∈ V)
2 p1val.t . . 3 1 = (1.‘𝐾)
3 fveq2 6888 . . . . . 6 (𝑘 = 𝐾 → (lub‘𝑘) = (lub‘𝐾))
4 p1val.u . . . . . 6 𝑈 = (lub‘𝐾)
53, 4eqtr4di 2791 . . . . 5 (𝑘 = 𝐾 → (lub‘𝑘) = 𝑈)
6 fveq2 6888 . . . . . 6 (𝑘 = 𝐾 → (Base‘𝑘) = (Base‘𝐾))
7 p1val.b . . . . . 6 𝐵 = (Base‘𝐾)
86, 7eqtr4di 2791 . . . . 5 (𝑘 = 𝐾 → (Base‘𝑘) = 𝐵)
95, 8fveq12d 6895 . . . 4 (𝑘 = 𝐾 → ((lub‘𝑘)‘(Base‘𝑘)) = (𝑈𝐵))
10 df-p1 18375 . . . 4 1. = (𝑘 ∈ V ↦ ((lub‘𝑘)‘(Base‘𝑘)))
11 fvex 6901 . . . 4 (𝑈𝐵) ∈ V
129, 10, 11fvmpt 6994 . . 3 (𝐾 ∈ V → (1.‘𝐾) = (𝑈𝐵))
132, 12eqtrid 2785 . 2 (𝐾 ∈ V → 1 = (𝑈𝐵))
141, 13syl 17 1 (𝐾𝑉1 = (𝑈𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1542  wcel 2107  Vcvv 3475  cfv 6540  Basecbs 17140  lubclub 18258  1.cp1 18373
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-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5298  ax-nul 5305  ax-pr 5426
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-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-iota 6492  df-fun 6542  df-fv 6548  df-p1 18375
This theorem is referenced by:  ple1  18379  clatp1cl  32125  xrsp1  32161  op1cl  37993
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