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Theorem bdayfo 31953
Description: The birthday function maps the surreals onto the ordinals. Alling's axiom (B). (Shortened proof on 2012-Apr-14, 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 7139 . . . 4 (𝑥 No → dom 𝑥 ∈ V)
21rgen 2951 . . 3 𝑥 No dom 𝑥 ∈ V
3 df-bday 31923 . . . 4 bday = (𝑥 No ↦ dom 𝑥)
43mptfng 6057 . . 3 (∀𝑥 No dom 𝑥 ∈ V ↔ bday Fn No )
52, 4mpbi 220 . 2 bday Fn No
63rnmpt 5403 . . 3 ran bday = {𝑦 ∣ ∃𝑥 No 𝑦 = dom 𝑥}
7 noxp1o 31941 . . . . . 6 (𝑦 ∈ On → (𝑦 × {1𝑜}) ∈ No )
8 1on 7612 . . . . . . . . . 10 1𝑜 ∈ On
98elexi 3244 . . . . . . . . 9 1𝑜 ∈ V
109snnz 4340 . . . . . . . 8 {1𝑜} ≠ ∅
11 dmxp 5376 . . . . . . . 8 ({1𝑜} ≠ ∅ → dom (𝑦 × {1𝑜}) = 𝑦)
1210, 11ax-mp 5 . . . . . . 7 dom (𝑦 × {1𝑜}) = 𝑦
1312eqcomi 2660 . . . . . 6 𝑦 = dom (𝑦 × {1𝑜})
14 dmeq 5356 . . . . . . . 8 (𝑥 = (𝑦 × {1𝑜}) → dom 𝑥 = dom (𝑦 × {1𝑜}))
1514eqeq2d 2661 . . . . . . 7 (𝑥 = (𝑦 × {1𝑜}) → (𝑦 = dom 𝑥𝑦 = dom (𝑦 × {1𝑜})))
1615rspcev 3340 . . . . . 6 (((𝑦 × {1𝑜}) ∈ No 𝑦 = dom (𝑦 × {1𝑜})) → ∃𝑥 No 𝑦 = dom 𝑥)
177, 13, 16sylancl 695 . . . . 5 (𝑦 ∈ On → ∃𝑥 No 𝑦 = dom 𝑥)
18 nodmon 31928 . . . . . . 7 (𝑥 No → dom 𝑥 ∈ On)
19 eleq1a 2725 . . . . . . 7 (dom 𝑥 ∈ On → (𝑦 = dom 𝑥𝑦 ∈ On))
2018, 19syl 17 . . . . . 6 (𝑥 No → (𝑦 = dom 𝑥𝑦 ∈ On))
2120rexlimiv 3056 . . . . 5 (∃𝑥 No 𝑦 = dom 𝑥𝑦 ∈ On)
2217, 21impbii 199 . . . 4 (𝑦 ∈ On ↔ ∃𝑥 No 𝑦 = dom 𝑥)
2322abbi2i 2767 . . 3 On = {𝑦 ∣ ∃𝑥 No 𝑦 = dom 𝑥}
246, 23eqtr4i 2676 . 2 ran bday = On
25 df-fo 5932 . 2 ( bday : No onto→On ↔ ( bday Fn No ∧ ran bday = On))
265, 24, 25mpbir2an 975 1 bday : No onto→On
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1523  wcel 2030  {cab 2637  wne 2823  wral 2941  wrex 2942  Vcvv 3231  c0 3948  {csn 4210   × cxp 5141  dom cdm 5143  ran crn 5144  Oncon0 5761   Fn wfn 5921  ontowfo 5924  1𝑜c1o 7598   No csur 31918   bday cbday 31920
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-ord 5764  df-on 5765  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-1o 7605  df-no 31921  df-bday 31923
This theorem is referenced by:  nodense  31967  bdayimaon  31968  nosupno  31974  nosupbday  31976  noetalem3  31990  noetalem4  31991  bdayfun  32013  bdayfn  32014  bdaydm  32015  bdayrn  32016  bdayelon  32017  noprc  32020
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