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Theorem bdayval 31045
 Description: The value of the birthday function within the surreals. (Contributed by Scott Fenton, 14-Jun-2011.)
Assertion
Ref Expression
bdayval (𝐴 No → ( bday 𝐴) = dom 𝐴)

Proof of Theorem bdayval
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 dmexg 6989 . 2 (𝐴 No → dom 𝐴 ∈ V)
2 dmeq 5246 . . 3 (𝑥 = 𝐴 → dom 𝑥 = dom 𝐴)
3 df-bday 31042 . . 3 bday = (𝑥 No ↦ dom 𝑥)
42, 3fvmptg 6189 . 2 ((𝐴 No ∧ dom 𝐴 ∈ V) → ( bday 𝐴) = dom 𝐴)
51, 4mpdan 699 1 (𝐴 No → ( bday 𝐴) = dom 𝐴)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1475   ∈ wcel 1977  Vcvv 3173  dom cdm 5038  ‘cfv 5804   No csur 31037   bday cbday 31039 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-8 1979  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  ax-un 6847 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-rn 5049  df-iota 5768  df-fun 5806  df-fv 5812  df-bday 31042 This theorem is referenced by:  nofnbday  31049  fvnobday  31081  nodenselem3  31082  nodenselem5  31084  nodense  31088  nobndlem3  31093  nofulllem3  31103
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