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Theorem nofnbday 30080
Description: A surreal is a function over its birthday. (Contributed by Scott Fenton, 16-Jun-2011.)
Assertion
Ref Expression
nofnbday  |-  ( A  e.  No  ->  A  Fn  ( bday `  A
) )

Proof of Theorem nofnbday
StepHypRef Expression
1 nofun 30077 . 2  |-  ( A  e.  No  ->  Fun  A )
2 bdayval 30076 . . 3  |-  ( A  e.  No  ->  ( bday `  A )  =  dom  A )
32eqcomd 2408 . 2  |-  ( A  e.  No  ->  dom  A  =  ( bday `  A
) )
4 df-fn 5526 . 2  |-  ( A  Fn  ( bday `  A
)  <->  ( Fun  A  /\  dom  A  =  (
bday `  A )
) )
51, 3, 4sylanbrc 662 1  |-  ( A  e.  No  ->  A  Fn  ( bday `  A
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1403    e. wcel 1840   dom cdm 4940   Fun wfun 5517    Fn wfn 5518   ` cfv 5523   Nocsur 30068   bdaycbday 30070
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1637  ax-4 1650  ax-5 1723  ax-6 1769  ax-7 1812  ax-8 1842  ax-9 1844  ax-10 1859  ax-11 1864  ax-12 1876  ax-13 2024  ax-ext 2378  ax-rep 4504  ax-sep 4514  ax-nul 4522  ax-pr 4627  ax-un 6528
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 974  df-tru 1406  df-ex 1632  df-nf 1636  df-sb 1762  df-eu 2240  df-mo 2241  df-clab 2386  df-cleq 2392  df-clel 2395  df-nfc 2550  df-ne 2598  df-ral 2756  df-rex 2757  df-reu 2758  df-rab 2760  df-v 3058  df-sbc 3275  df-csb 3371  df-dif 3414  df-un 3416  df-in 3418  df-ss 3425  df-nul 3736  df-if 3883  df-sn 3970  df-pr 3972  df-op 3976  df-uni 4189  df-iun 4270  df-br 4393  df-opab 4451  df-mpt 4452  df-id 4735  df-xp 4946  df-rel 4947  df-cnv 4948  df-co 4949  df-dm 4950  df-rn 4951  df-res 4952  df-ima 4953  df-iota 5487  df-fun 5525  df-fn 5526  df-f 5527  df-f1 5528  df-fo 5529  df-f1o 5530  df-fv 5531  df-no 30071  df-bday 30073
This theorem is referenced by:  nodenselem4  30112  nodenselem6  30114  nodenselem8  30116
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