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Theorem axdenselem1 23503
Description: Lemma for axdense 23511. 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 23471 . 2  |-  ( A  e.  No  ->  Fun  A )
2 bdayval 23470 . . 3  |-  ( A  e.  No  ->  ( bday `  A )  =  dom  A )
32eqcomd 2258 . 2  |-  ( A  e.  No  ->  dom  A  =  ( bday `  A
) )
4 df-fn 4603 . 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 4580   Fun wfun 4586    Fn wfn 4587   ` cfv 4592   Nocsur 23462   bdaycbday 23464
This theorem is referenced by:  axdenselem4  23506  axdenselem6  23508  axdenselem8  23510
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 1926  ax-ext 2234  ax-rep 4028  ax-sep 4038  ax-nul 4046  ax-pr 4108  ax-un 4403
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 1883  df-eu 2118  df-mo 2119  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-ne 2414  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2516  df-v 2729  df-sbc 2922  df-csb 3010  df-dif 3081  df-un 3083  df-in 3085  df-ss 3089  df-nul 3363  df-if 3471  df-sn 3550  df-pr 3551  df-op 3553  df-uni 3728  df-iun 3805  df-br 3921  df-opab 3975  df-mpt 3976  df-id 4202  df-xp 4594  df-rel 4595  df-cnv 4596  df-co 4597  df-dm 4598  df-rn 4599  df-res 4600  df-ima 4601  df-fun 4602  df-fn 4603  df-f 4604  df-f1 4605  df-fo 4606  df-f1o 4607  df-fv 4608  df-no 23465  df-bday 23467
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