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Theorem nofnbday 31789
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 31786 . 2 (𝐴 No → Fun 𝐴)
2 bdayval 31785 . . 3 (𝐴 No → ( bday 𝐴) = dom 𝐴)
32eqcomd 2627 . 2 (𝐴 No → dom 𝐴 = ( bday 𝐴))
4 df-fn 5889 . 2 (𝐴 Fn ( bday 𝐴) ↔ (Fun 𝐴 ∧ dom 𝐴 = ( bday 𝐴)))
51, 3, 4sylanbrc 698 1 (𝐴 No 𝐴 Fn ( bday 𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1482  wcel 1989  dom cdm 5112  Fun wfun 5880   Fn wfn 5881  cfv 5886   No csur 31777   bday cbday 31779
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1721  ax-4 1736  ax-5 1838  ax-6 1887  ax-7 1934  ax-8 1991  ax-9 1998  ax-10 2018  ax-11 2033  ax-12 2046  ax-13 2245  ax-ext 2601  ax-rep 4769  ax-sep 4779  ax-nul 4787  ax-pr 4904  ax-un 6946
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1485  df-ex 1704  df-nf 1709  df-sb 1880  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2752  df-ne 2794  df-ral 2916  df-rex 2917  df-reu 2918  df-rab 2920  df-v 3200  df-sbc 3434  df-csb 3532  df-dif 3575  df-un 3577  df-in 3579  df-ss 3586  df-nul 3914  df-if 4085  df-sn 4176  df-pr 4178  df-op 4182  df-uni 4435  df-iun 4520  df-br 4652  df-opab 4711  df-mpt 4728  df-id 5022  df-xp 5118  df-rel 5119  df-cnv 5120  df-co 5121  df-dm 5122  df-rn 5123  df-res 5124  df-ima 5125  df-iota 5849  df-fun 5888  df-fn 5889  df-f 5890  df-f1 5891  df-fo 5892  df-f1o 5893  df-fv 5894  df-no 31780  df-bday 31782
This theorem is referenced by:  nodenselem8  31825
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