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Theorem bdayval 33160
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 7589 . 2 (𝐴 No → dom 𝐴 ∈ V)
2 dmeq 5746 . . 3 (𝑥 = 𝐴 → dom 𝑥 = dom 𝐴)
3 df-bday 33157 . . 3 bday = (𝑥 No ↦ dom 𝑥)
42, 3fvmptg 6740 . 2 ((𝐴 No ∧ dom 𝐴 ∈ V) → ( bday 𝐴) = dom 𝐴)
51, 4mpdan 685 1 (𝐴 No → ( bday 𝐴) = dom 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1537  wcel 2114  Vcvv 3473  dom cdm 5529  cfv 6329   No csur 33152   bday cbday 33154
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2792  ax-sep 5177  ax-nul 5184  ax-pr 5304  ax-un 7437
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2799  df-cleq 2813  df-clel 2891  df-nfc 2959  df-ral 3130  df-rex 3131  df-rab 3134  df-v 3475  df-sbc 3752  df-dif 3915  df-un 3917  df-in 3919  df-ss 3928  df-nul 4268  df-if 4442  df-sn 4542  df-pr 4544  df-op 4548  df-uni 4813  df-br 5041  df-opab 5103  df-mpt 5121  df-id 5434  df-xp 5535  df-rel 5536  df-cnv 5537  df-co 5538  df-dm 5539  df-rn 5540  df-iota 6288  df-fun 6331  df-fv 6337  df-bday 33157
This theorem is referenced by:  nofnbday  33164  fvnobday  33188  nodenselem5  33197  nodense  33201  nosupno  33208  nosupbday  33210  noetalem3  33224  noetalem4  33225
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