Theorem bdayfo 28064

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⟶ontoOn

Proof of Theorem bdayfo

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		dmexg 7937	. . . 4 ⊢ (𝑥 ∈ No → dom 𝑥 ∈ V)
2	1	rgen 3082	. . 3 ⊢ ∀𝑥 ∈ No dom 𝑥 ∈ V
3		df-bday 28032	. . . 4 ⊢ bday = (𝑥 ∈ No ↦ dom 𝑥)
4	3	mptfng 6700	. . 3 ⊢ (∀𝑥 ∈ No dom 𝑥 ∈ V ↔ bday Fn No)
5	2, 4	mpbi 233	. 2 ⊢ bday Fn No
6	3	rnmpt 5946	. . 3 ⊢ ran bday = {𝑦 ∣ ∃𝑥 ∈ No 𝑦 = dom 𝑥}
7		noxp1o 28050	. . . . . 6 ⊢ (𝑦 ∈ On → (𝑦 × {1o}) ∈ No)
8		1oex 8514	. . . . . . . . 9 ⊢ 1o ∈ V
9	8	snnz 4737	. . . . . . . 8 ⊢ {1o} ≠ ∅
10		dmxp 5918	. . . . . . . 8 ⊢ ({1o} ≠ ∅ → dom (𝑦 × {1o}) = 𝑦)
11	9, 10	ax-mp 5	. . . . . . 7 ⊢ dom (𝑦 × {1o}) = 𝑦
12	11	eqcomi 2772	. . . . . 6 ⊢ 𝑦 = dom (𝑦 × {1o})
13		dmeq 5892	. . . . . . 7 ⊢ (𝑥 = (𝑦 × {1o}) → dom 𝑥 = dom (𝑦 × {1o}))
14	13	rspceeqv 3602	. . . . . 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 28037	. . . . . . 7 ⊢ (𝑥 ∈ No → dom 𝑥 ∈ On)
17		eleq1a 2859	. . . . . . 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 3160	. . . . 5 ⊢ (∃𝑥 ∈ No 𝑦 = dom 𝑥 → 𝑦 ∈ On)
20	15, 19	impbii 212	. . . 4 ⊢ (𝑦 ∈ On ↔ ∃𝑥 ∈ No 𝑦 = dom 𝑥)
21	20	eqabi 2899	. . 3 ⊢ On = {𝑦 ∣ ∃𝑥 ∈ No 𝑦 = dom 𝑥}
22	6, 21	eqtr4i 2789	. 2 ⊢ ran bday = On
23		df-fo 6564	. 2 ⊢ (bday:No⟶ontoOn ↔ (bday Fn No ∧ ran bday = On))
24	5, 22, 23	mpbir2an 724	1 ⊢ bday:No⟶ontoOn

This proof depends on syntax axioms:   → wi 4   = wceq 1570   ∈ wcel 2147  {cab 2741   ≠ wne 2959  ∀wral 3080  ∃wrex 3090  Vcvv 3454  ∅c0 4281  {csn 4588   × cxp 5653  dom cdm 5655  ran crn 5656  Oncon0 6380   Fn wfn 6553  ⟶ontowfo 6556  1_oc1o 8497  Nocsur 28027  bdaycbday 28029

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-sep 5252  ax-pow 5331  ax-pr 5395  ax-un 7775

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 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-br 5107  df-opab 5171  df-mpt 5190  df-id 5550  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-suc 6386  df-fun 6560  df-fn 6561  df-f 6562  df-fo 6564  df-1o 8504  df-no 28030  df-bday 28032

This theorem is used by:  nodense  28079  bdayimaon  28080  nosupno  28090  nosupbday  28092  noinfno  28105  noinfbday  28107  noetasuplem4  28123  noetainflem4  28127  bdayfun  28163  bdayfn  28164  bdaydmOLD  28166  bdayrn  28167  bdayon  28168  noprc  28172