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Theorem bdayfo 30909
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 6865 . . . 4 (𝑥 No → dom 𝑥 ∈ V)
21rgen 2810 . . 3 𝑥 No dom 𝑥 ∈ V
3 df-bday 30877 . . . 4 bday = (𝑥 No ↦ dom 𝑥)
43mptfng 5817 . . 3 (∀𝑥 No dom 𝑥 ∈ V ↔ bday Fn No )
52, 4mpbi 218 . 2 bday Fn No
63rnmpt 5183 . . 3 ran bday = {𝑦 ∣ ∃𝑥 No 𝑦 = dom 𝑥}
7 noxp1o 30898 . . . . . 6 (𝑦 ∈ On → (𝑦 × {1𝑜}) ∈ No )
8 1on 7330 . . . . . . . . . 10 1𝑜 ∈ On
98elexi 3090 . . . . . . . . 9 1𝑜 ∈ V
109snnz 4155 . . . . . . . 8 {1𝑜} ≠ ∅
11 dmxp 5156 . . . . . . . 8 ({1𝑜} ≠ ∅ → dom (𝑦 × {1𝑜}) = 𝑦)
1210, 11ax-mp 5 . . . . . . 7 dom (𝑦 × {1𝑜}) = 𝑦
1312eqcomi 2523 . . . . . 6 𝑦 = dom (𝑦 × {1𝑜})
14 dmeq 5137 . . . . . . . 8 (𝑥 = (𝑦 × {1𝑜}) → dom 𝑥 = dom (𝑦 × {1𝑜}))
1514eqeq2d 2524 . . . . . . 7 (𝑥 = (𝑦 × {1𝑜}) → (𝑦 = dom 𝑥𝑦 = dom (𝑦 × {1𝑜})))
1615rspcev 3186 . . . . . 6 (((𝑦 × {1𝑜}) ∈ No 𝑦 = dom (𝑦 × {1𝑜})) → ∃𝑥 No 𝑦 = dom 𝑥)
177, 13, 16sylancl 692 . . . . 5 (𝑦 ∈ On → ∃𝑥 No 𝑦 = dom 𝑥)
18 nodmon 30882 . . . . . . 7 (𝑥 No → dom 𝑥 ∈ On)
19 eleq1a 2587 . . . . . . 7 (dom 𝑥 ∈ On → (𝑦 = dom 𝑥𝑦 ∈ On))
2018, 19syl 17 . . . . . 6 (𝑥 No → (𝑦 = dom 𝑥𝑦 ∈ On))
2120rexlimiv 2913 . . . . 5 (∃𝑥 No 𝑦 = dom 𝑥𝑦 ∈ On)
2217, 21impbii 197 . . . 4 (𝑦 ∈ On ↔ ∃𝑥 No 𝑦 = dom 𝑥)
2322abbi2i 2629 . . 3 On = {𝑦 ∣ ∃𝑥 No 𝑦 = dom 𝑥}
246, 23eqtr4i 2539 . 2 ran bday = On
25 df-fo 5695 . 2 ( bday : No onto→On ↔ ( bday Fn No ∧ ran bday = On))
265, 24, 25mpbir2an 956 1 bday : No onto→On
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1474  wcel 1938  {cab 2500  wne 2684  wral 2800  wrex 2801  Vcvv 3077  c0 3777  {csn 4028   × cxp 4930  dom cdm 4932  ran crn 4933  Oncon0 5530   Fn wfn 5684  ontowfo 5687  1𝑜c1o 7316   No csur 30872   bday cbday 30874
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1700  ax-4 1713  ax-5 1793  ax-6 1838  ax-7 1885  ax-8 1940  ax-9 1947  ax-10 1966  ax-11 1971  ax-12 1983  ax-13 2137  ax-ext 2494  ax-rep 4597  ax-sep 4607  ax-nul 4616  ax-pr 4732  ax-un 6723
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3or 1031  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1699  df-sb 1831  df-eu 2366  df-mo 2367  df-clab 2501  df-cleq 2507  df-clel 2510  df-nfc 2644  df-ne 2686  df-ral 2805  df-rex 2806  df-reu 2807  df-rab 2809  df-v 3079  df-sbc 3307  df-csb 3404  df-dif 3447  df-un 3449  df-in 3451  df-ss 3458  df-pss 3460  df-nul 3778  df-if 3940  df-pw 4013  df-sn 4029  df-pr 4031  df-tp 4033  df-op 4035  df-uni 4271  df-iun 4355  df-br 4482  df-opab 4542  df-mpt 4543  df-tr 4579  df-eprel 4843  df-id 4847  df-po 4853  df-so 4854  df-fr 4891  df-we 4893  df-xp 4938  df-rel 4939  df-cnv 4940  df-co 4941  df-dm 4942  df-rn 4943  df-res 4944  df-ima 4945  df-ord 5533  df-on 5534  df-suc 5536  df-iota 5653  df-fun 5691  df-fn 5692  df-f 5693  df-f1 5694  df-fo 5695  df-f1o 5696  df-fv 5697  df-1o 7323  df-no 30875  df-bday 30877
This theorem is referenced by:  bdayfun  30910  bdayrn  30911  bdaydm  30912  noprc  30915
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