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Theorem axdenselem1 23669
Description: Lemma for axdense 23677. A surreal is a function over its birthday. (Contributed by Scott Fenton, 16-Jun-2011.)
Assertion
Ref Expression
axdenselem1  |-  ( A  e.  No  ->  A  Fn  ( bday `  A
) )

Proof of Theorem axdenselem1
StepHypRef Expression
1 nofun 23637 . 2  |-  ( A  e.  No  ->  Fun  A )
2 bdayval 23636 . . 3  |-  ( A  e.  No  ->  ( bday `  A )  =  dom  A )
32eqcomd 2261 . 2  |-  ( A  e.  No  ->  dom  A  =  ( bday `  A
) )
4 df-fn 4649 . 2  |-  ( A  Fn  ( bday `  A
)  <->  ( Fun  A  /\  dom  A  =  (
bday `  A )
) )
51, 3, 4sylanbrc 648 1  |-  ( A  e.  No  ->  A  Fn  ( bday `  A
) )
Colors of variables: wff set class
Syntax hints:    -> wi 6    = wceq 1619    e. wcel 1621   dom cdm 4626   Fun wfun 4632    Fn wfn 4633   ` cfv 4638   Nocsur 23628   bdaycbday 23630
This theorem is referenced by:  axdenselem4  23672  axdenselem6  23674  axdenselem8  23676
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2237  ax-rep 4071  ax-sep 4081  ax-nul 4089  ax-pr 4152  ax-un 4449
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-eu 2121  df-mo 2122  df-clab 2243  df-cleq 2249  df-clel 2252  df-nfc 2381  df-ne 2421  df-ral 2520  df-rex 2521  df-reu 2522  df-rab 2523  df-v 2742  df-sbc 2936  df-csb 3024  df-dif 3097  df-un 3099  df-in 3101  df-ss 3108  df-nul 3398  df-if 3507  df-sn 3587  df-pr 3588  df-op 3590  df-uni 3769  df-iun 3848  df-br 3964  df-opab 4018  df-mpt 4019  df-id 4246  df-xp 4640  df-rel 4641  df-cnv 4642  df-co 4643  df-dm 4644  df-rn 4645  df-res 4646  df-ima 4647  df-fun 4648  df-fn 4649  df-f 4650  df-f1 4651  df-fo 4652  df-f1o 4653  df-fv 4654  df-no 23631  df-bday 23633
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