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Theorem nofnbday 32142
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 32139 . 2 (𝐴 No → Fun 𝐴)
2 bdayval 32138 . . 3 (𝐴 No → ( bday 𝐴) = dom 𝐴)
32eqcomd 2777 . 2 (𝐴 No → dom 𝐴 = ( bday 𝐴))
4 df-fn 6034 . 2 (𝐴 Fn ( bday 𝐴) ↔ (Fun 𝐴 ∧ dom 𝐴 = ( bday 𝐴)))
51, 3, 4sylanbrc 572 1 (𝐴 No 𝐴 Fn ( bday 𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1631  wcel 2145  dom cdm 5249  Fun wfun 6025   Fn wfn 6026  cfv 6031   No csur 32130   bday cbday 32132
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4904  ax-sep 4915  ax-nul 4923  ax-pr 5034  ax-un 7096
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-iun 4656  df-br 4787  df-opab 4847  df-mpt 4864  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-no 32133  df-bday 32135
This theorem is referenced by:  nodenselem8  32178
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