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Theorem nofnbday 33592
Description: A surreal is a function over its birthday. (Contributed by Scott Fenton, 16-Jun-2011.)
Assertion
Ref Expression
nofnbday (𝐴 No 𝐴 Fn ( bday 𝐴))

Proof of Theorem nofnbday
StepHypRef Expression
1 nofun 33589 . 2 (𝐴 No → Fun 𝐴)
2 bdayval 33588 . . 3 (𝐴 No → ( bday 𝐴) = dom 𝐴)
32eqcomd 2743 . 2 (𝐴 No → dom 𝐴 = ( bday 𝐴))
4 df-fn 6383 . 2 (𝐴 Fn ( bday 𝐴) ↔ (Fun 𝐴 ∧ dom 𝐴 = ( bday 𝐴)))
51, 3, 4sylanbrc 586 1 (𝐴 No 𝐴 Fn ( bday 𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1543  wcel 2110  dom cdm 5551  Fun wfun 6374   Fn wfn 6375  cfv 6380   No csur 33580   bday cbday 33582
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708  ax-rep 5179  ax-sep 5192  ax-nul 5199  ax-pr 5322  ax-un 7523
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-ne 2941  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3410  df-sbc 3695  df-csb 3812  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-nul 4238  df-if 4440  df-sn 4542  df-pr 4544  df-op 4548  df-uni 4820  df-iun 4906  df-br 5054  df-opab 5116  df-mpt 5136  df-id 5455  df-xp 5557  df-rel 5558  df-cnv 5559  df-co 5560  df-dm 5561  df-rn 5562  df-res 5563  df-ima 5564  df-iota 6338  df-fun 6382  df-fn 6383  df-f 6384  df-f1 6385  df-fo 6386  df-f1o 6387  df-fv 6388  df-no 33583  df-bday 33585
This theorem is referenced by:  nodenselem8  33631
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