Theorem nofnbday 27624

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 27621	. 2 ⊢ (𝐴 ∈ No → Fun 𝐴)
2		bdayval 27620	. . 3 ⊢ (𝐴 ∈ No → ( bday ‘𝐴) = dom 𝐴)
3	2	eqcomd 2743	. 2 ⊢ (𝐴 ∈ No → dom 𝐴 = ( bday ‘𝐴))
4		df-fn 6496	. 2 ⊢ (𝐴 Fn ( bday ‘𝐴) ↔ (Fun 𝐴 ∧ dom 𝐴 = ( bday ‘𝐴)))
5	1, 3, 4	sylanbrc 584	1 ⊢ (𝐴 ∈ No → 𝐴 Fn ( bday ‘𝐴))

Syntax hints:   → wi 4   = wceq 1542   ∈ wcel 2114  dom cdm 5625  Fun wfun 6487   Fn wfn 6488  ‘cfv 6493   No csur 27611   bday cbday 27613

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5242  ax-nul 5252  ax-pow 5311  ax-pr 5378  ax-un 7682

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ral 3053  df-rex 3062  df-rab 3401  df-v 3443  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4287  df-if 4481  df-pw 4557  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-br 5100  df-opab 5162  df-mpt 5181  df-id 5520  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-iota 6449  df-fun 6495  df-fn 6496  df-f 6497  df-fv 6501  df-no 27614  df-bday 27616

This theorem is referenced by:  nodenselem8  27663