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Theorem bdayval 27019
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 7844 . 2 (𝐴 No → dom 𝐴 ∈ V)
2 dmeq 5863 . . 3 (𝑥 = 𝐴 → dom 𝑥 = dom 𝐴)
3 df-bday 27016 . . 3 bday = (𝑥 No ↦ dom 𝑥)
42, 3fvmptg 6950 . 2 ((𝐴 No ∧ dom 𝐴 ∈ V) → ( bday 𝐴) = dom 𝐴)
51, 4mpdan 686 1 (𝐴 No → ( bday 𝐴) = dom 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1542  wcel 2107  Vcvv 3447  dom cdm 5637  cfv 6500   No csur 27011   bday cbday 27013
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 5260  ax-nul 5267  ax-pr 5388  ax-un 7676
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-ral 3062  df-rex 3071  df-rab 3407  df-v 3449  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4287  df-if 4491  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-br 5110  df-opab 5172  df-mpt 5193  df-id 5535  df-xp 5643  df-rel 5644  df-cnv 5645  df-co 5646  df-dm 5647  df-rn 5648  df-iota 6452  df-fun 6502  df-fv 6508  df-bday 27016
This theorem is referenced by:  nofnbday  27023  fvnobday  27049  nodenselem5  27059  nodense  27063  nosupno  27074  nosupbday  27076  noinfno  27089  noinfbday  27091  noetasuplem4  27107  noetainflem4  27111  onnobdayg  41794  bdaybndex  41795  bdaybndbday  41796
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