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Theorem bdayfo 32709
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 7428 . . . 4 (𝑥 No → dom 𝑥 ∈ V)
21rgen 3098 . . 3 𝑥 No dom 𝑥 ∈ V
3 df-bday 32679 . . . 4 bday = (𝑥 No ↦ dom 𝑥)
43mptfng 6317 . . 3 (∀𝑥 No dom 𝑥 ∈ V ↔ bday Fn No )
52, 4mpbi 222 . 2 bday Fn No
63rnmpt 5670 . . 3 ran bday = {𝑦 ∣ ∃𝑥 No 𝑦 = dom 𝑥}
7 noxp1o 32697 . . . . . 6 (𝑦 ∈ On → (𝑦 × {1o}) ∈ No )
8 1oex 7913 . . . . . . . . 9 1o ∈ V
98snnz 4585 . . . . . . . 8 {1o} ≠ ∅
10 dmxp 5642 . . . . . . . 8 ({1o} ≠ ∅ → dom (𝑦 × {1o}) = 𝑦)
119, 10ax-mp 5 . . . . . . 7 dom (𝑦 × {1o}) = 𝑦
1211eqcomi 2787 . . . . . 6 𝑦 = dom (𝑦 × {1o})
13 dmeq 5622 . . . . . . 7 (𝑥 = (𝑦 × {1o}) → dom 𝑥 = dom (𝑦 × {1o}))
1413rspceeqv 3553 . . . . . 6 (((𝑦 × {1o}) ∈ No 𝑦 = dom (𝑦 × {1o})) → ∃𝑥 No 𝑦 = dom 𝑥)
157, 12, 14sylancl 577 . . . . 5 (𝑦 ∈ On → ∃𝑥 No 𝑦 = dom 𝑥)
16 nodmon 32684 . . . . . . 7 (𝑥 No → dom 𝑥 ∈ On)
17 eleq1a 2861 . . . . . . 7 (dom 𝑥 ∈ On → (𝑦 = dom 𝑥𝑦 ∈ On))
1816, 17syl 17 . . . . . 6 (𝑥 No → (𝑦 = dom 𝑥𝑦 ∈ On))
1918rexlimiv 3225 . . . . 5 (∃𝑥 No 𝑦 = dom 𝑥𝑦 ∈ On)
2015, 19impbii 201 . . . 4 (𝑦 ∈ On ↔ ∃𝑥 No 𝑦 = dom 𝑥)
2120abbi2i 2905 . . 3 On = {𝑦 ∣ ∃𝑥 No 𝑦 = dom 𝑥}
226, 21eqtr4i 2805 . 2 ran bday = On
23 df-fo 6194 . 2 ( bday : No onto→On ↔ ( bday Fn No ∧ ran bday = On))
245, 22, 23mpbir2an 698 1 bday : No onto→On
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1507  wcel 2050  {cab 2758  wne 2967  wral 3088  wrex 3089  Vcvv 3415  c0 4178  {csn 4441   × cxp 5405  dom cdm 5407  ran crn 5408  Oncon0 6029   Fn wfn 6183  ontowfo 6186  1oc1o 7898   No csur 32674   bday cbday 32676
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-13 2301  ax-ext 2750  ax-rep 5049  ax-sep 5060  ax-nul 5067  ax-pr 5186  ax-un 7279
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3or 1069  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2016  df-mo 2547  df-eu 2584  df-clab 2759  df-cleq 2771  df-clel 2846  df-nfc 2918  df-ne 2968  df-ral 3093  df-rex 3094  df-reu 3095  df-rab 3097  df-v 3417  df-sbc 3682  df-csb 3787  df-dif 3832  df-un 3834  df-in 3836  df-ss 3843  df-pss 3845  df-nul 4179  df-if 4351  df-pw 4424  df-sn 4442  df-pr 4444  df-tp 4446  df-op 4448  df-uni 4713  df-iun 4794  df-br 4930  df-opab 4992  df-mpt 5009  df-tr 5031  df-id 5312  df-eprel 5317  df-po 5326  df-so 5327  df-fr 5366  df-we 5368  df-xp 5413  df-rel 5414  df-cnv 5415  df-co 5416  df-dm 5417  df-rn 5418  df-res 5419  df-ima 5420  df-ord 6032  df-on 6033  df-suc 6035  df-iota 6152  df-fun 6190  df-fn 6191  df-f 6192  df-f1 6193  df-fo 6194  df-f1o 6195  df-fv 6196  df-1o 7905  df-no 32677  df-bday 32679
This theorem is referenced by:  nodense  32723  bdayimaon  32724  nosupno  32730  nosupbday  32732  noetalem3  32746  noetalem4  32747  bdayfun  32769  bdayfn  32770  bdaydm  32771  bdayrn  32772  bdayelon  32773  noprc  32776
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