Theorem bdayfo 27852

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 7896	. . . 4 ⊢ (𝑥 ∈ No → dom 𝑥 ∈ V)
2	1	rgen 3080	. . 3 ⊢ ∀𝑥 ∈ No dom 𝑥 ∈ V
3		df-bday 27820	. . . 4 ⊢ bday = (𝑥 ∈ No ↦ dom 𝑥)
4	3	mptfng 6674	. . 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 27838	. . . . . 6 ⊢ (𝑦 ∈ On → (𝑦 × {1o}) ∈ No )
8		1oex 8461	. . . . . . . . 9 ⊢ 1o ∈ V
9	8	snnz 4741	. . . . . . . 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 2771	. . . . . 6 ⊢ 𝑦 = dom (𝑦 × {1o})
13		dmeq 5892	. . . . . . 7 ⊢ (𝑥 = (𝑦 × {1o}) → dom 𝑥 = dom (𝑦 × {1o}))
14	13	rspceeqv 3603	. . . . . 6 ⊢ (((𝑦 × {1o}) ∈ No ∧ 𝑦 = dom (𝑦 × {1o})) → ∃𝑥 ∈ No 𝑦 = dom 𝑥)
15	7, 12, 14	sylancl 597	. . . . 5 ⊢ (𝑦 ∈ On → ∃𝑥 ∈ No 𝑦 = dom 𝑥)
16		nodmon 27825	. . . . . . 7 ⊢ (𝑥 ∈ No → dom 𝑥 ∈ On)
17		eleq1a 2857	. . . . . . 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 3158	. . . . 5 ⊢ (∃𝑥 ∈ No 𝑦 = dom 𝑥 → 𝑦 ∈ On)
20	15, 19	impbii 212	. . . 4 ⊢ (𝑦 ∈ On ↔ ∃𝑥 ∈ No 𝑦 = dom 𝑥)
21	20	eqabi 2897	. . 3 ⊢ On = {𝑦 ∣ ∃𝑥 ∈ No 𝑦 = dom 𝑥}
22	6, 21	eqtr4i 2788	. 2 ⊢ ran bday = On
23		df-fo 6542	. 2 ⊢ ( bday : No –onto→On ↔ ( bday Fn No ∧ ran bday = On))
24	5, 22, 23	mpbir2an 723	1 ⊢ bday : No –onto→On

This proof depends on syntax axioms:   → wi 4   = wceq 1569   ∈ wcel 2142  {cab 2740   ≠ wne 2957  ∀wral 3078  ∃wrex 3088  Vcvv 3454  ∅c0 4285  {csn 4588   × cxp 5658  dom cdm 5660  ran crn 5661  Oncon0 6360   Fn wfn 6531  –onto→wfo 6534  1_oc1o 8444   No csur 27815   bday cbday 27817

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734  ax-sep 5256  ax-pow 5335  ax-pr 5403  ax-un 7734

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1104  df-tru 1572  df-fal 1582  df-ex 1809  df-nf 1813  df-sb 2096  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3079  df-rex 3089  df-rab 3416  df-v 3456  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4487  df-pw 4563  df-sn 4589  df-pr 4591  df-op 4595  df-uni 4872  df-br 5109  df-opab 5173  df-mpt 5192  df-id 5555  df-xp 5666  df-rel 5667  df-cnv 5668  df-co 5669  df-dm 5670  df-rn 5671  df-suc 6366  df-fun 6538  df-fn 6539  df-f 6540  df-fo 6542  df-1o 8451  df-no 27818  df-bday 27820

This theorem is used by:  nodense  27867  bdayimaon  27868  nosupno  27878  nosupbday  27880  noinfno  27893  noinfbday  27895  noetasuplem4  27911  noetainflem4  27915  bdayfun  27951  bdayfn  27952  bdaydmOLD  27954  bdayrn  27955  bdayon  27956  noprc  27960  noeta2  27965