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Theorem bdayfo 27551
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 –ontoβ†’On

Proof of Theorem bdayfo
Dummy variables π‘₯ 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 dmexg 7888 . . . 4 (π‘₯ ∈ No β†’ dom π‘₯ ∈ V)
21rgen 3055 . . 3 βˆ€π‘₯ ∈ No dom π‘₯ ∈ V
3 df-bday 27519 . . . 4 bday = (π‘₯ ∈ No ↦ dom π‘₯)
43mptfng 6680 . . 3 (βˆ€π‘₯ ∈ No dom π‘₯ ∈ V ↔ bday Fn No )
52, 4mpbi 229 . 2 bday Fn No
63rnmpt 5945 . . 3 ran bday = {𝑦 ∣ βˆƒπ‘₯ ∈ No 𝑦 = dom π‘₯}
7 noxp1o 27537 . . . . . 6 (𝑦 ∈ On β†’ (𝑦 Γ— {1o}) ∈ No )
8 1oex 8472 . . . . . . . . 9 1o ∈ V
98snnz 4773 . . . . . . . 8 {1o} β‰  βˆ…
10 dmxp 5919 . . . . . . . 8 ({1o} β‰  βˆ… β†’ dom (𝑦 Γ— {1o}) = 𝑦)
119, 10ax-mp 5 . . . . . . 7 dom (𝑦 Γ— {1o}) = 𝑦
1211eqcomi 2733 . . . . . 6 𝑦 = dom (𝑦 Γ— {1o})
13 dmeq 5894 . . . . . . 7 (π‘₯ = (𝑦 Γ— {1o}) β†’ dom π‘₯ = dom (𝑦 Γ— {1o}))
1413rspceeqv 3626 . . . . . 6 (((𝑦 Γ— {1o}) ∈ No ∧ 𝑦 = dom (𝑦 Γ— {1o})) β†’ βˆƒπ‘₯ ∈ No 𝑦 = dom π‘₯)
157, 12, 14sylancl 585 . . . . 5 (𝑦 ∈ On β†’ βˆƒπ‘₯ ∈ No 𝑦 = dom π‘₯)
16 nodmon 27524 . . . . . . 7 (π‘₯ ∈ No β†’ dom π‘₯ ∈ On)
17 eleq1a 2820 . . . . . . 7 (dom π‘₯ ∈ On β†’ (𝑦 = dom π‘₯ β†’ 𝑦 ∈ On))
1816, 17syl 17 . . . . . 6 (π‘₯ ∈ No β†’ (𝑦 = dom π‘₯ β†’ 𝑦 ∈ On))
1918rexlimiv 3140 . . . . 5 (βˆƒπ‘₯ ∈ No 𝑦 = dom π‘₯ β†’ 𝑦 ∈ On)
2015, 19impbii 208 . . . 4 (𝑦 ∈ On ↔ βˆƒπ‘₯ ∈ No 𝑦 = dom π‘₯)
2120eqabi 2861 . . 3 On = {𝑦 ∣ βˆƒπ‘₯ ∈ No 𝑦 = dom π‘₯}
226, 21eqtr4i 2755 . 2 ran bday = On
23 df-fo 6540 . 2 ( bday : No –ontoβ†’On ↔ ( bday Fn No ∧ ran bday = On))
245, 22, 23mpbir2an 708 1 bday : No –ontoβ†’On
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   = wceq 1533   ∈ wcel 2098  {cab 2701   β‰  wne 2932  βˆ€wral 3053  βˆƒwrex 3062  Vcvv 3466  βˆ…c0 4315  {csn 4621   Γ— cxp 5665  dom cdm 5667  ran crn 5668  Oncon0 6355   Fn wfn 6529  β€“ontoβ†’wfo 6532  1oc1o 8455   No csur 27514   bday cbday 27516
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-rep 5276  ax-sep 5290  ax-nul 5297  ax-pr 5418  ax-un 7719
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-ral 3054  df-rex 3063  df-reu 3369  df-rab 3425  df-v 3468  df-sbc 3771  df-csb 3887  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4316  df-if 4522  df-sn 4622  df-pr 4624  df-op 4628  df-uni 4901  df-iun 4990  df-br 5140  df-opab 5202  df-mpt 5223  df-id 5565  df-xp 5673  df-rel 5674  df-cnv 5675  df-co 5676  df-dm 5677  df-rn 5678  df-res 5679  df-ima 5680  df-suc 6361  df-iota 6486  df-fun 6536  df-fn 6537  df-f 6538  df-f1 6539  df-fo 6540  df-f1o 6541  df-fv 6542  df-1o 8462  df-no 27517  df-bday 27519
This theorem is referenced by:  nodense  27566  bdayimaon  27567  nosupno  27577  nosupbday  27579  noinfno  27592  noinfbday  27594  noetasuplem4  27610  noetainflem4  27614  bdayfun  27646  bdayfn  27647  bdaydm  27648  bdayrn  27649  bdayelon  27650  noprc  27653  noeta2  27658
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