Theorem bdayfo 31953

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.

Step	Hyp	Ref	Expression
1		dmexg 7139	. . . 4 ⊢ (𝑥 ∈ No → dom 𝑥 ∈ V)
2	1	rgen 2951	. . 3 ⊢ ∀𝑥 ∈ No dom 𝑥 ∈ V
3		df-bday 31923	. . . 4 ⊢ bday = (𝑥 ∈ No ↦ dom 𝑥)
4	3	mptfng 6057	. . 3 ⊢ (∀𝑥 ∈ No dom 𝑥 ∈ V ↔ bday Fn No )
5	2, 4	mpbi 220	. 2 ⊢ bday Fn No
6	3	rnmpt 5403	. . 3 ⊢ ran bday = {𝑦 ∣ ∃𝑥 ∈ No 𝑦 = dom 𝑥}
7		noxp1o 31941	. . . . . 6 ⊢ (𝑦 ∈ On → (𝑦 × {1𝑜}) ∈ No )
8		1on 7612	. . . . . . . . . 10 ⊢ 1𝑜 ∈ On
9	8	elexi 3244	. . . . . . . . 9 ⊢ 1𝑜 ∈ V
10	9	snnz 4340	. . . . . . . 8 ⊢ {1𝑜} ≠ ∅
11		dmxp 5376	. . . . . . . 8 ⊢ ({1𝑜} ≠ ∅ → dom (𝑦 × {1𝑜}) = 𝑦)
12	10, 11	ax-mp 5	. . . . . . 7 ⊢ dom (𝑦 × {1𝑜}) = 𝑦
13	12	eqcomi 2660	. . . . . 6 ⊢ 𝑦 = dom (𝑦 × {1𝑜})
14		dmeq 5356	. . . . . . . 8 ⊢ (𝑥 = (𝑦 × {1𝑜}) → dom 𝑥 = dom (𝑦 × {1𝑜}))
15	14	eqeq2d 2661	. . . . . . 7 ⊢ (𝑥 = (𝑦 × {1𝑜}) → (𝑦 = dom 𝑥 ↔ 𝑦 = dom (𝑦 × {1𝑜})))
16	15	rspcev 3340	. . . . . 6 ⊢ (((𝑦 × {1𝑜}) ∈ No ∧ 𝑦 = dom (𝑦 × {1𝑜})) → ∃𝑥 ∈ No 𝑦 = dom 𝑥)
17	7, 13, 16	sylancl 695	. . . . 5 ⊢ (𝑦 ∈ On → ∃𝑥 ∈ No 𝑦 = dom 𝑥)
18		nodmon 31928	. . . . . . 7 ⊢ (𝑥 ∈ No → dom 𝑥 ∈ On)
19		eleq1a 2725	. . . . . . 7 ⊢ (dom 𝑥 ∈ On → (𝑦 = dom 𝑥 → 𝑦 ∈ On))
20	18, 19	syl 17	. . . . . 6 ⊢ (𝑥 ∈ No → (𝑦 = dom 𝑥 → 𝑦 ∈ On))
21	20	rexlimiv 3056	. . . . 5 ⊢ (∃𝑥 ∈ No 𝑦 = dom 𝑥 → 𝑦 ∈ On)
22	17, 21	impbii 199	. . . 4 ⊢ (𝑦 ∈ On ↔ ∃𝑥 ∈ No 𝑦 = dom 𝑥)
23	22	abbi2i 2767	. . 3 ⊢ On = {𝑦 ∣ ∃𝑥 ∈ No 𝑦 = dom 𝑥}
24	6, 23	eqtr4i 2676	. 2 ⊢ ran bday = On
25		df-fo 5932	. 2 ⊢ ( bday : No –onto→On ↔ ( bday Fn No ∧ ran bday = On))
26	5, 24, 25	mpbir2an 975	1 ⊢ bday : No –onto→On

Syntax hints:   → wi 4   = wceq 1523   ∈ wcel 2030  {cab 2637   ≠ wne 2823  ∀wral 2941  ∃wrex 2942  Vcvv 3231  ∅c0 3948  {csn 4210   × cxp 5141  dom cdm 5143  ran crn 5144  Oncon0 5761   Fn wfn 5921  –onto→wfo 5924  1_𝑜c1o 7598   No csur 31918   bday cbday 31920

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pr 4936  ax-un 6991

This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-ord 5764  df-on 5765  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-1o 7605  df-no 31921  df-bday 31923

This theorem is referenced by:  nodense  31967  bdayimaon  31968  nosupno  31974  nosupbday  31976  noetalem3  31990  noetalem4  31991  bdayfun  32013  bdayfn  32014  bdaydm  32015  bdayrn  32016  bdayelon  32017  noprc  32020