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Theorem axdenselem1 23737
Description: Lemma for axdense 23745. 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 23705 . 2  |-  ( A  e.  No  ->  Fun  A )
2 bdayval 23704 . . 3  |-  ( A  e.  No  ->  ( bday `  A )  =  dom  A )
32eqcomd 2290 . 2  |-  ( A  e.  No  ->  dom  A  =  ( bday `  A
) )
4 df-fn 5225 . 2  |-  ( A  Fn  ( bday `  A
)  <->  ( Fun  A  /\  dom  A  =  (
bday `  A )
) )
51, 3, 4sylanbrc 647 1  |-  ( A  e.  No  ->  A  Fn  ( bday `  A
) )
Colors of variables: wff set class
Syntax hints:    -> wi 6    = wceq 1624    e. wcel 1685   dom cdm 4689   Fun wfun 5216    Fn wfn 5217   ` cfv 5222   Nocsur 23696   bdaycbday 23698
This theorem is referenced by:  axdenselem4  23740  axdenselem6  23742  axdenselem8  23744
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1534  ax-5 1545  ax-17 1604  ax-9 1637  ax-8 1645  ax-13 1687  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1868  ax-ext 2266  ax-rep 4133  ax-sep 4143  ax-nul 4151  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 938  df-tru 1312  df-ex 1530  df-nf 1533  df-sb 1632  df-eu 2149  df-mo 2150  df-clab 2272  df-cleq 2278  df-clel 2281  df-nfc 2410  df-ne 2450  df-ral 2550  df-rex 2551  df-reu 2552  df-rab 2554  df-v 2792  df-sbc 2994  df-csb 3084  df-dif 3157  df-un 3159  df-in 3161  df-ss 3168  df-nul 3458  df-if 3568  df-sn 3648  df-pr 3649  df-op 3651  df-uni 3830  df-iun 3909  df-br 4026  df-opab 4080  df-mpt 4081  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-fun 5224  df-fn 5225  df-f 5226  df-f1 5227  df-fo 5228  df-f1o 5229  df-fv 5230  df-no 23699  df-bday 23701
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