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Theorem bdayfo 31074
 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 6989 . . . 4 (𝑥 No → dom 𝑥 ∈ V)
21rgen 2906 . . 3 𝑥 No dom 𝑥 ∈ V
3 df-bday 31042 . . . 4 bday = (𝑥 No ↦ dom 𝑥)
43mptfng 5932 . . 3 (∀𝑥 No dom 𝑥 ∈ V ↔ bday Fn No )
52, 4mpbi 219 . 2 bday Fn No
63rnmpt 5292 . . 3 ran bday = {𝑦 ∣ ∃𝑥 No 𝑦 = dom 𝑥}
7 noxp1o 31063 . . . . . 6 (𝑦 ∈ On → (𝑦 × {1𝑜}) ∈ No )
8 1on 7454 . . . . . . . . . 10 1𝑜 ∈ On
98elexi 3186 . . . . . . . . 9 1𝑜 ∈ V
109snnz 4252 . . . . . . . 8 {1𝑜} ≠ ∅
11 dmxp 5265 . . . . . . . 8 ({1𝑜} ≠ ∅ → dom (𝑦 × {1𝑜}) = 𝑦)
1210, 11ax-mp 5 . . . . . . 7 dom (𝑦 × {1𝑜}) = 𝑦
1312eqcomi 2619 . . . . . 6 𝑦 = dom (𝑦 × {1𝑜})
14 dmeq 5246 . . . . . . . 8 (𝑥 = (𝑦 × {1𝑜}) → dom 𝑥 = dom (𝑦 × {1𝑜}))
1514eqeq2d 2620 . . . . . . 7 (𝑥 = (𝑦 × {1𝑜}) → (𝑦 = dom 𝑥𝑦 = dom (𝑦 × {1𝑜})))
1615rspcev 3282 . . . . . 6 (((𝑦 × {1𝑜}) ∈ No 𝑦 = dom (𝑦 × {1𝑜})) → ∃𝑥 No 𝑦 = dom 𝑥)
177, 13, 16sylancl 693 . . . . 5 (𝑦 ∈ On → ∃𝑥 No 𝑦 = dom 𝑥)
18 nodmon 31047 . . . . . . 7 (𝑥 No → dom 𝑥 ∈ On)
19 eleq1a 2683 . . . . . . 7 (dom 𝑥 ∈ On → (𝑦 = dom 𝑥𝑦 ∈ On))
2018, 19syl 17 . . . . . 6 (𝑥 No → (𝑦 = dom 𝑥𝑦 ∈ On))
2120rexlimiv 3009 . . . . 5 (∃𝑥 No 𝑦 = dom 𝑥𝑦 ∈ On)
2217, 21impbii 198 . . . 4 (𝑦 ∈ On ↔ ∃𝑥 No 𝑦 = dom 𝑥)
2322abbi2i 2725 . . 3 On = {𝑦 ∣ ∃𝑥 No 𝑦 = dom 𝑥}
246, 23eqtr4i 2635 . 2 ran bday = On
25 df-fo 5810 . 2 ( bday : No onto→On ↔ ( bday Fn No ∧ ran bday = On))
265, 24, 25mpbir2an 957 1 bday : No onto→On
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1475   ∈ wcel 1977  {cab 2596   ≠ wne 2780  ∀wral 2896  ∃wrex 2897  Vcvv 3173  ∅c0 3874  {csn 4125   × cxp 5036  dom cdm 5038  ran crn 5039  Oncon0 5640   Fn wfn 5799  –onto→wfo 5802  1𝑜c1o 7440   No csur 31037   bday cbday 31039 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-ord 5643  df-on 5644  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-1o 7447  df-no 31040  df-bday 31042 This theorem is referenced by:  bdayfun  31075  bdayrn  31076  bdaydm  31077  noprc  31080
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