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Theorem bdayval 29582
Description: The value of the birthday function within the surreals. (Contributed by Scott Fenton, 14-Jun-2011.)
Assertion
Ref Expression
bdayval  |-  ( A  e.  No  ->  ( bday `  A )  =  dom  A )

Proof of Theorem bdayval
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 dmexg 6730 . 2  |-  ( A  e.  No  ->  dom  A  e.  _V )
2 dmeq 5213 . . 3  |-  ( x  =  A  ->  dom  x  =  dom  A )
3 df-bday 29579 . . 3  |-  bday  =  ( x  e.  No  |->  dom  x )
42, 3fvmptg 5954 . 2  |-  ( ( A  e.  No  /\  dom  A  e.  _V )  ->  ( bday `  A
)  =  dom  A
)
51, 4mpdan 668 1  |-  ( A  e.  No  ->  ( bday `  A )  =  dom  A )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1395    e. wcel 1819   _Vcvv 3109   dom cdm 5008   ` cfv 5594   Nocsur 29574   bdaycbday 29576
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pr 4695  ax-un 6591
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-sbc 3328  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-br 4457  df-opab 4516  df-mpt 4517  df-id 4804  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-iota 5557  df-fun 5596  df-fv 5602  df-bday 29579
This theorem is referenced by:  nofnbday  29586  fvnobday  29616  nodenselem3  29617  nodenselem5  29619  nodense  29623  nobndlem3  29628  nofulllem3  29638
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