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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
StepHypRef Expression
1 nofun 32677 . 2 (𝐴 No → Fun 𝐴)
2 bdayval 32676 . . 3 (𝐴 No → ( bday 𝐴) = dom 𝐴)
32eqcomd 2778 . 2 (𝐴 No → dom 𝐴 = ( bday 𝐴))
4 df-fn 6185 . 2 (𝐴 Fn ( bday 𝐴) ↔ (Fun 𝐴 ∧ dom 𝐴 = ( bday 𝐴)))
51, 3, 4sylanbrc 575 1 (𝐴 No 𝐴 Fn ( bday 𝐴))
Colors of variables: wff setvar class
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
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