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Theorem bdayval 25595
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 5122 . 2  |-  ( A  e.  No  ->  dom  A  e.  _V )
2 dmeq 5062 . . 3  |-  ( x  =  A  ->  dom  x  =  dom  A )
3 df-bday 25592 . . 3  |-  bday  =  ( x  e.  No  |->  dom  x )
42, 3fvmptg 5796 . 2  |-  ( ( A  e.  No  /\  dom  A  e.  _V )  ->  ( bday `  A
)  =  dom  A
)
51, 4mpdan 650 1  |-  ( A  e.  No  ->  ( bday `  A )  =  dom  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1652    e. wcel 1725   _Vcvv 2948   dom cdm 4870   ` cfv 5446   Nocsur 25587   bdaycbday 25589
This theorem is referenced by:  nofnbday  25599  fvnobday  25629  nodenselem3  25630  nodenselem5  25632  nodense  25636  nobndlem3  25641  nofulllem3  25651
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pr 4395  ax-un 4693
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-iota 5410  df-fun 5448  df-fv 5454  df-bday 25592
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