Theorem nofnbday 27618

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

Step	Hyp	Ref	Expression
1		nofun 27615	. 2 ⊢ (𝐴 ∈ No → Fun 𝐴)
2		bdayval 27614	. . 3 ⊢ (𝐴 ∈ No → ( bday ‘𝐴) = dom 𝐴)
3	2	eqcomd 2740	. 2 ⊢ (𝐴 ∈ No → dom 𝐴 = ( bday ‘𝐴))
4		df-fn 6493	. 2 ⊢ (𝐴 Fn ( bday ‘𝐴) ↔ (Fun 𝐴 ∧ dom 𝐴 = ( bday ‘𝐴)))
5	1, 3, 4	sylanbrc 583	1 ⊢ (𝐴 ∈ No → 𝐴 Fn ( bday ‘𝐴))

Syntax hints:   → wi 4   = wceq 1541   ∈ wcel 2113  dom cdm 5622  Fun wfun 6484   Fn wfn 6485  ‘cfv 6490   No csur 27605   bday cbday 27607

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2706  ax-sep 5239  ax-nul 5249  ax-pow 5308  ax-pr 5375  ax-un 7678

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2809  df-nfc 2883  df-ral 3050  df-rex 3059  df-rab 3398  df-v 3440  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4284  df-if 4478  df-pw 4554  df-sn 4579  df-pr 4581  df-op 4585  df-uni 4862  df-br 5097  df-opab 5159  df-mpt 5178  df-id 5517  df-xp 5628  df-rel 5629  df-cnv 5630  df-co 5631  df-dm 5632  df-rn 5633  df-iota 6446  df-fun 6492  df-fn 6493  df-f 6494  df-fv 6498  df-no 27608  df-bday 27610

This theorem is referenced by:  nodenselem8  27657