Theorem nofnbday 26910

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 26907	. 2 ⊢ (𝐴 ∈ No → Fun 𝐴)
2		bdayval 26906	. . 3 ⊢ (𝐴 ∈ No → ( bday ‘𝐴) = dom 𝐴)
3	2	eqcomd 2743	. 2 ⊢ (𝐴 ∈ No → dom 𝐴 = ( bday ‘𝐴))
4		df-fn 6491	. 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 1541   ∈ wcel 2106  dom cdm 5627  Fun wfun 6482   Fn wfn 6483  ‘cfv 6488   No csur 26898   bday cbday 26900

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2708  ax-rep 5237  ax-sep 5251  ax-nul 5258  ax-pr 5379  ax-un 7659

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2887  df-ne 2942  df-ral 3063  df-rex 3072  df-reu 3352  df-rab 3406  df-v 3445  df-sbc 3735  df-csb 3851  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4278  df-if 4482  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4861  df-iun 4951  df-br 5101  df-opab 5163  df-mpt 5184  df-id 5525  df-xp 5633  df-rel 5634  df-cnv 5635  df-co 5636  df-dm 5637  df-rn 5638  df-res 5639  df-ima 5640  df-iota 6440  df-fun 6490  df-fn 6491  df-f 6492  df-f1 6493  df-fo 6494  df-f1o 6495  df-fv 6496  df-no 26901  df-bday 26903

This theorem is referenced by:  nodenselem8  26949