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Theorem bdayval 31502
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 7044 . 2 (𝐴 No → dom 𝐴 ∈ V)
2 dmeq 5284 . . 3 (𝑥 = 𝐴 → dom 𝑥 = dom 𝐴)
3 df-bday 31499 . . 3 bday = (𝑥 No ↦ dom 𝑥)
42, 3fvmptg 6237 . 2 ((𝐴 No ∧ dom 𝐴 ∈ V) → ( bday 𝐴) = dom 𝐴)
51, 4mpdan 701 1 (𝐴 No → ( bday 𝐴) = dom 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1480  wcel 1987  Vcvv 3186  dom cdm 5074  cfv 5847   No csur 31494   bday cbday 31496
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pr 4867  ax-un 6902
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-sbc 3418  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-br 4614  df-opab 4674  df-mpt 4675  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-iota 5810  df-fun 5849  df-fv 5855  df-bday 31499
This theorem is referenced by:  nofnbday  31506  fvnobday  31545  nodenselem3  31546  nodenselem5  31548  nodense  31552  nobndlem3  31557  nosino  31575
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