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Theorem bdayfo 27707
Description: The birthday function maps the surreals onto the ordinals. Axiom B of [Alling] p. 184. (Proof shortened on 14-Apr-2012 by SF). (Contributed by Scott Fenton, 11-Jun-2011.)
Assertion
Ref Expression
bdayfo bday : No onto→On

Proof of Theorem bdayfo
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 dmexg 7914 . . . 4 (𝑥 No → dom 𝑥 ∈ V)
21rgen 3053 . . 3 𝑥 No dom 𝑥 ∈ V
3 df-bday 27674 . . . 4 bday = (𝑥 No ↦ dom 𝑥)
43mptfng 6700 . . 3 (∀𝑥 No dom 𝑥 ∈ V ↔ bday Fn No )
52, 4mpbi 229 . 2 bday Fn No
63rnmpt 5961 . . 3 ran bday = {𝑦 ∣ ∃𝑥 No 𝑦 = dom 𝑥}
7 noxp1o 27693 . . . . . 6 (𝑦 ∈ On → (𝑦 × {1o}) ∈ No )
8 1oex 8506 . . . . . . . . 9 1o ∈ V
98snnz 4785 . . . . . . . 8 {1o} ≠ ∅
10 dmxp 5935 . . . . . . . 8 ({1o} ≠ ∅ → dom (𝑦 × {1o}) = 𝑦)
119, 10ax-mp 5 . . . . . . 7 dom (𝑦 × {1o}) = 𝑦
1211eqcomi 2735 . . . . . 6 𝑦 = dom (𝑦 × {1o})
13 dmeq 5910 . . . . . . 7 (𝑥 = (𝑦 × {1o}) → dom 𝑥 = dom (𝑦 × {1o}))
1413rspceeqv 3630 . . . . . 6 (((𝑦 × {1o}) ∈ No 𝑦 = dom (𝑦 × {1o})) → ∃𝑥 No 𝑦 = dom 𝑥)
157, 12, 14sylancl 584 . . . . 5 (𝑦 ∈ On → ∃𝑥 No 𝑦 = dom 𝑥)
16 nodmon 27680 . . . . . . 7 (𝑥 No → dom 𝑥 ∈ On)
17 eleq1a 2821 . . . . . . 7 (dom 𝑥 ∈ On → (𝑦 = dom 𝑥𝑦 ∈ On))
1816, 17syl 17 . . . . . 6 (𝑥 No → (𝑦 = dom 𝑥𝑦 ∈ On))
1918rexlimiv 3138 . . . . 5 (∃𝑥 No 𝑦 = dom 𝑥𝑦 ∈ On)
2015, 19impbii 208 . . . 4 (𝑦 ∈ On ↔ ∃𝑥 No 𝑦 = dom 𝑥)
2120eqabi 2862 . . 3 On = {𝑦 ∣ ∃𝑥 No 𝑦 = dom 𝑥}
226, 21eqtr4i 2757 . 2 ran bday = On
23 df-fo 6560 . 2 ( bday : No onto→On ↔ ( bday Fn No ∧ ran bday = On))
245, 22, 23mpbir2an 709 1 bday : No onto→On
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1534  wcel 2099  {cab 2703  wne 2930  wral 3051  wrex 3060  Vcvv 3462  c0 4325  {csn 4633   × cxp 5680  dom cdm 5682  ran crn 5683  Oncon0 6376   Fn wfn 6549  ontowfo 6552  1oc1o 8489   No csur 27669   bday cbday 27671
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-sep 5304  ax-nul 5311  ax-pow 5369  ax-pr 5433  ax-un 7746
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ne 2931  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3464  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4326  df-if 4534  df-pw 4609  df-sn 4634  df-pr 4636  df-op 4640  df-uni 4914  df-br 5154  df-opab 5216  df-mpt 5237  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-suc 6382  df-fun 6556  df-fn 6557  df-f 6558  df-fo 6560  df-1o 8496  df-no 27672  df-bday 27674
This theorem is referenced by:  nodense  27722  bdayimaon  27723  nosupno  27733  nosupbday  27735  noinfno  27748  noinfbday  27750  noetasuplem4  27766  noetainflem4  27770  bdayfun  27802  bdayfn  27803  bdaydm  27804  bdayrn  27805  bdayelon  27806  noprc  27809  noeta2  27814
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