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Theorem bdayfo 33296
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 7598 . . . 4 (𝑥 No → dom 𝑥 ∈ V)
21rgen 3119 . . 3 𝑥 No dom 𝑥 ∈ V
3 df-bday 33266 . . . 4 bday = (𝑥 No ↦ dom 𝑥)
43mptfng 6463 . . 3 (∀𝑥 No dom 𝑥 ∈ V ↔ bday Fn No )
52, 4mpbi 233 . 2 bday Fn No
63rnmpt 5795 . . 3 ran bday = {𝑦 ∣ ∃𝑥 No 𝑦 = dom 𝑥}
7 noxp1o 33284 . . . . . 6 (𝑦 ∈ On → (𝑦 × {1o}) ∈ No )
8 1oex 8097 . . . . . . . . 9 1o ∈ V
98snnz 4675 . . . . . . . 8 {1o} ≠ ∅
10 dmxp 5767 . . . . . . . 8 ({1o} ≠ ∅ → dom (𝑦 × {1o}) = 𝑦)
119, 10ax-mp 5 . . . . . . 7 dom (𝑦 × {1o}) = 𝑦
1211eqcomi 2810 . . . . . 6 𝑦 = dom (𝑦 × {1o})
13 dmeq 5740 . . . . . . 7 (𝑥 = (𝑦 × {1o}) → dom 𝑥 = dom (𝑦 × {1o}))
1413rspceeqv 3589 . . . . . 6 (((𝑦 × {1o}) ∈ No 𝑦 = dom (𝑦 × {1o})) → ∃𝑥 No 𝑦 = dom 𝑥)
157, 12, 14sylancl 589 . . . . 5 (𝑦 ∈ On → ∃𝑥 No 𝑦 = dom 𝑥)
16 nodmon 33271 . . . . . . 7 (𝑥 No → dom 𝑥 ∈ On)
17 eleq1a 2888 . . . . . . 7 (dom 𝑥 ∈ On → (𝑦 = dom 𝑥𝑦 ∈ On))
1816, 17syl 17 . . . . . 6 (𝑥 No → (𝑦 = dom 𝑥𝑦 ∈ On))
1918rexlimiv 3242 . . . . 5 (∃𝑥 No 𝑦 = dom 𝑥𝑦 ∈ On)
2015, 19impbii 212 . . . 4 (𝑦 ∈ On ↔ ∃𝑥 No 𝑦 = dom 𝑥)
2120abbi2i 2932 . . 3 On = {𝑦 ∣ ∃𝑥 No 𝑦 = dom 𝑥}
226, 21eqtr4i 2827 . 2 ran bday = On
23 df-fo 6334 . 2 ( bday : No onto→On ↔ ( bday Fn No ∧ ran bday = On))
245, 22, 23mpbir2an 710 1 bday : No onto→On
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1538  wcel 2112  {cab 2779  wne 2990  wral 3109  wrex 3110  Vcvv 3444  c0 4246  {csn 4528   × cxp 5521  dom cdm 5523  ran crn 5524  Oncon0 6163   Fn wfn 6323  ontowfo 6326  1oc1o 8082   No csur 33261   bday cbday 33263
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-rep 5157  ax-sep 5170  ax-nul 5177  ax-pr 5298  ax-un 7445
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-ral 3114  df-rex 3115  df-reu 3116  df-rab 3118  df-v 3446  df-sbc 3724  df-csb 3832  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-pss 3903  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-tp 4533  df-op 4535  df-uni 4804  df-iun 4886  df-br 5034  df-opab 5096  df-mpt 5114  df-tr 5140  df-id 5428  df-eprel 5433  df-po 5442  df-so 5443  df-fr 5482  df-we 5484  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-ord 6166  df-on 6167  df-suc 6169  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-1o 8089  df-no 33264  df-bday 33266
This theorem is referenced by:  nodense  33310  bdayimaon  33311  nosupno  33317  nosupbday  33319  noetalem3  33333  noetalem4  33334  bdayfun  33356  bdayfn  33357  bdaydm  33358  bdayrn  33359  bdayelon  33360  noprc  33363
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