Theorem nofnbday 32680

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 32677	. 2 ⊢ (𝐴 ∈ No → Fun 𝐴)
2		bdayval 32676	. . 3 ⊢ (𝐴 ∈ No → ( bday ‘𝐴) = dom 𝐴)
3	2	eqcomd 2778	. 2 ⊢ (𝐴 ∈ No → dom 𝐴 = ( bday ‘𝐴))
4		df-fn 6185	. 2 ⊢ (𝐴 Fn ( bday ‘𝐴) ↔ (Fun 𝐴 ∧ dom 𝐴 = ( bday ‘𝐴)))
5	1, 3, 4	sylanbrc 575	1 ⊢ (𝐴 ∈ No → 𝐴 Fn ( bday ‘𝐴))

Syntax hints:   → wi 4   = wceq 1507   ∈ wcel 2050  dom cdm 5401  Fun wfun 6176   Fn wfn 6177  ‘cfv 6182   No csur 32668   bday cbday 32670

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-13 2301  ax-ext 2744  ax-rep 5043  ax-sep 5054  ax-nul 5061  ax-pr 5180  ax-un 7273

This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2016  df-mo 2547  df-eu 2584  df-clab 2753  df-cleq 2765  df-clel 2840  df-nfc 2912  df-ne 2962  df-ral 3087  df-rex 3088  df-reu 3089  df-rab 3091  df-v 3411  df-sbc 3676  df-csb 3781  df-dif 3826  df-un 3828  df-in 3830  df-ss 3837  df-nul 4173  df-if 4345  df-sn 4436  df-pr 4438  df-op 4442  df-uni 4707  df-iun 4788  df-br 4924  df-opab 4986  df-mpt 5003  df-id 5306  df-xp 5407  df-rel 5408  df-cnv 5409  df-co 5410  df-dm 5411  df-rn 5412  df-res 5413  df-ima 5414  df-iota 6146  df-fun 6184  df-fn 6185  df-f 6186  df-f1 6187  df-fo 6188  df-f1o 6189  df-fv 6190  df-no 32671  df-bday 32673

This theorem is referenced by:  nodenselem8  32716