Theorem bdayfo 27878

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 –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 7907	. . . 4 ⊢ (𝑥 ∈ No → dom 𝑥 ∈ V)
2	1	rgen 3084	. . 3 ⊢ ∀𝑥 ∈ No dom 𝑥 ∈ V
3		df-bday 27846	. . . 4 ⊢ bday = (𝑥 ∈ No ↦ dom 𝑥)
4	3	mptfng 6681	. . 3 ⊢ (∀𝑥 ∈ No dom 𝑥 ∈ V ↔ bday Fn No )
5	2, 4	mpbi 233	. 2 ⊢ bday Fn No
6	3	rnmpt 5952	. . 3 ⊢ ran bday = {𝑦 ∣ ∃𝑥 ∈ No 𝑦 = dom 𝑥}
7		noxp1o 27864	. . . . . 6 ⊢ (𝑦 ∈ On → (𝑦 × {1o}) ∈ No )
8		1oex 8472	. . . . . . . . 9 ⊢ 1o ∈ V
9	8	snnz 4747	. . . . . . . 8 ⊢ {1o} ≠ ∅
10		dmxp 5924	. . . . . . . 8 ⊢ ({1o} ≠ ∅ → dom (𝑦 × {1o}) = 𝑦)
11	9, 10	ax-mp 5	. . . . . . 7 ⊢ dom (𝑦 × {1o}) = 𝑦
12	11	eqcomi 2775	. . . . . 6 ⊢ 𝑦 = dom (𝑦 × {1o})
13		dmeq 5898	. . . . . . 7 ⊢ (𝑥 = (𝑦 × {1o}) → dom 𝑥 = dom (𝑦 × {1o}))
14	13	rspceeqv 3607	. . . . . 6 ⊢ (((𝑦 × {1o}) ∈ No ∧ 𝑦 = dom (𝑦 × {1o})) → ∃𝑥 ∈ No 𝑦 = dom 𝑥)
15	7, 12, 14	sylancl 598	. . . . 5 ⊢ (𝑦 ∈ On → ∃𝑥 ∈ No 𝑦 = dom 𝑥)
16		nodmon 27851	. . . . . . 7 ⊢ (𝑥 ∈ No → dom 𝑥 ∈ On)
17		eleq1a 2861	. . . . . . 7 ⊢ (dom 𝑥 ∈ On → (𝑦 = dom 𝑥 → 𝑦 ∈ On))
18	16, 17	syl 18	. . . . . 6 ⊢ (𝑥 ∈ No → (𝑦 = dom 𝑥 → 𝑦 ∈ On))
19	18	rexlimiv 3162	. . . . 5 ⊢ (∃𝑥 ∈ No 𝑦 = dom 𝑥 → 𝑦 ∈ On)
20	15, 19	impbii 212	. . . 4 ⊢ (𝑦 ∈ On ↔ ∃𝑥 ∈ No 𝑦 = dom 𝑥)
21	20	eqabi 2901	. . 3 ⊢ On = {𝑦 ∣ ∃𝑥 ∈ No 𝑦 = dom 𝑥}
22	6, 21	eqtr4i 2792	. 2 ⊢ ran bday = On
23		df-fo 6549	. 2 ⊢ ( bday : No –onto→On ↔ ( bday Fn No ∧ ran bday = On))
24	5, 22, 23	mpbir2an 724	1 ⊢ bday : No –onto→On

This proof depends on syntax axioms:   → wi 4   = wceq 1570   ∈ wcel 2146  {cab 2744   ≠ wne 2961  ∀wral 3082  ∃wrex 3092  Vcvv 3458  ∅c0 4289  {csn 4594   × cxp 5664  dom cdm 5666  ran crn 5667  Oncon0 6367   Fn wfn 6538  –onto→wfo 6541  1_oc1o 8455   No csur 27841   bday cbday 27843

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2738  ax-sep 5262  ax-pow 5341  ax-pr 5409  ax-un 7745

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2570  df-eu 2600  df-clab 2745  df-cleq 2758  df-clel 2841  df-nfc 2915  df-ne 2962  df-ral 3083  df-rex 3093  df-rab 3420  df-v 3460  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4290  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4878  df-br 5115  df-opab 5179  df-mpt 5198  df-id 5561  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-suc 6373  df-fun 6545  df-fn 6546  df-f 6547  df-fo 6549  df-1o 8462  df-no 27844  df-bday 27846

This theorem is used by:  nodense  27893  bdayimaon  27894  nosupno  27904  nosupbday  27906  noinfno  27919  noinfbday  27921  noetasuplem4  27937  noetainflem4  27941  bdayfun  27977  bdayfn  27978  bdaydmOLD  27980  bdayrn  27981  bdayon  27982  noprc  27986  noeta2  27991