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Theorem bdayval 32078
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 7250 . 2 (𝐴 No → dom 𝐴 ∈ V)
2 dmeq 5467 . . 3 (𝑥 = 𝐴 → dom 𝑥 = dom 𝐴)
3 df-bday 32075 . . 3 bday = (𝑥 No ↦ dom 𝑥)
42, 3fvmptg 6430 . 2 ((𝐴 No ∧ dom 𝐴 ∈ V) → ( bday 𝐴) = dom 𝐴)
51, 4mpdan 705 1 (𝐴 No → ( bday 𝐴) = dom 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1620  wcel 2127  Vcvv 3328  dom cdm 5254  cfv 6037   No csur 32070   bday cbday 32072
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1859  ax-4 1874  ax-5 1976  ax-6 2042  ax-7 2078  ax-8 2129  ax-9 2136  ax-10 2156  ax-11 2171  ax-12 2184  ax-13 2379  ax-ext 2728  ax-sep 4921  ax-nul 4929  ax-pr 5043  ax-un 7102
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1623  df-ex 1842  df-nf 1847  df-sb 2035  df-eu 2599  df-mo 2600  df-clab 2735  df-cleq 2741  df-clel 2744  df-nfc 2879  df-ral 3043  df-rex 3044  df-rab 3047  df-v 3330  df-sbc 3565  df-dif 3706  df-un 3708  df-in 3710  df-ss 3717  df-nul 4047  df-if 4219  df-sn 4310  df-pr 4312  df-op 4316  df-uni 4577  df-br 4793  df-opab 4853  df-mpt 4870  df-id 5162  df-xp 5260  df-rel 5261  df-cnv 5262  df-co 5263  df-dm 5264  df-rn 5265  df-iota 6000  df-fun 6039  df-fv 6045  df-bday 32075
This theorem is referenced by:  nofnbday  32082  fvnobday  32106  nodenselem5  32115  nodense  32119  nosupno  32126  nosupbday  32128  noetalem3  32142  noetalem4  32143
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