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Theorem bdayfo 27048
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 7844 . . . 4 (π‘₯ ∈ No β†’ dom π‘₯ ∈ V)
21rgen 3063 . . 3 βˆ€π‘₯ ∈ No dom π‘₯ ∈ V
3 df-bday 27016 . . . 4 bday = (π‘₯ ∈ No ↦ dom π‘₯)
43mptfng 6644 . . 3 (βˆ€π‘₯ ∈ No dom π‘₯ ∈ V ↔ bday Fn No )
52, 4mpbi 229 . 2 bday Fn No
63rnmpt 5914 . . 3 ran bday = {𝑦 ∣ βˆƒπ‘₯ ∈ No 𝑦 = dom π‘₯}
7 noxp1o 27034 . . . . . 6 (𝑦 ∈ On β†’ (𝑦 Γ— {1o}) ∈ No )
8 1oex 8426 . . . . . . . . 9 1o ∈ V
98snnz 4741 . . . . . . . 8 {1o} β‰  βˆ…
10 dmxp 5888 . . . . . . . 8 ({1o} β‰  βˆ… β†’ dom (𝑦 Γ— {1o}) = 𝑦)
119, 10ax-mp 5 . . . . . . 7 dom (𝑦 Γ— {1o}) = 𝑦
1211eqcomi 2742 . . . . . 6 𝑦 = dom (𝑦 Γ— {1o})
13 dmeq 5863 . . . . . . 7 (π‘₯ = (𝑦 Γ— {1o}) β†’ dom π‘₯ = dom (𝑦 Γ— {1o}))
1413rspceeqv 3599 . . . . . 6 (((𝑦 Γ— {1o}) ∈ No ∧ 𝑦 = dom (𝑦 Γ— {1o})) β†’ βˆƒπ‘₯ ∈ No 𝑦 = dom π‘₯)
157, 12, 14sylancl 587 . . . . 5 (𝑦 ∈ On β†’ βˆƒπ‘₯ ∈ No 𝑦 = dom π‘₯)
16 nodmon 27021 . . . . . . 7 (π‘₯ ∈ No β†’ dom π‘₯ ∈ On)
17 eleq1a 2829 . . . . . . 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 3142 . . . . 5 (βˆƒπ‘₯ ∈ No 𝑦 = dom π‘₯ β†’ 𝑦 ∈ On)
2015, 19impbii 208 . . . 4 (𝑦 ∈ On ↔ βˆƒπ‘₯ ∈ No 𝑦 = dom π‘₯)
2120abbi2i 2870 . . 3 On = {𝑦 ∣ βˆƒπ‘₯ ∈ No 𝑦 = dom π‘₯}
226, 21eqtr4i 2764 . 2 ran bday = On
23 df-fo 6506 . 2 ( bday : No –ontoβ†’On ↔ ( bday Fn No ∧ ran bday = On))
245, 22, 23mpbir2an 710 1 bday : No –ontoβ†’On
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   = wceq 1542   ∈ wcel 2107  {cab 2710   β‰  wne 2940  βˆ€wral 3061  βˆƒwrex 3070  Vcvv 3447  βˆ…c0 4286  {csn 4590   Γ— cxp 5635  dom cdm 5637  ran crn 5638  Oncon0 6321   Fn wfn 6495  β€“ontoβ†’wfo 6498  1oc1o 8409   No csur 27011   bday cbday 27013
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5246  ax-sep 5260  ax-nul 5267  ax-pr 5388  ax-un 7676
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3353  df-rab 3407  df-v 3449  df-sbc 3744  df-csb 3860  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4287  df-if 4491  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-iun 4960  df-br 5110  df-opab 5172  df-mpt 5193  df-id 5535  df-xp 5643  df-rel 5644  df-cnv 5645  df-co 5646  df-dm 5647  df-rn 5648  df-res 5649  df-ima 5650  df-suc 6327  df-iota 6452  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-1o 8416  df-no 27014  df-bday 27016
This theorem is referenced by:  nodense  27063  bdayimaon  27064  nosupno  27074  nosupbday  27076  noinfno  27089  noinfbday  27091  noetasuplem4  27107  noetainflem4  27111  bdayfun  27141  bdayfn  27142  bdaydm  27143  bdayrn  27144  bdayelon  27145  noprc  27148  noeta2  27153
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