Theorem bdayfo 28039

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 7913	. . . 4 ⊢ (𝑥 ∈ No → dom 𝑥 ∈ V)
2	1	rgen 3078	. . 3 ⊢ ∀𝑥 ∈ No dom 𝑥 ∈ V
3		df-bday 28007	. . . 4 ⊢ bday = (𝑥 ∈ No ↦ dom 𝑥)
4	3	mptfng 6678	. . 3 ⊢ (∀𝑥 ∈ No dom 𝑥 ∈ V ↔ bday Fn No)
5	2, 4	mpbi 233	. 2 ⊢ bday Fn No
6	3	rnmpt 5938	. . 3 ⊢ ran bday = {𝑦 ∣ ∃𝑥 ∈ No 𝑦 = dom 𝑥}
7		noxp1o 28025	. . . . . 6 ⊢ (𝑦 ∈ On → (𝑦 × {1o}) ∈ No)
8		1oex 8489	. . . . . . . . 9 ⊢ 1o ∈ V
9	8	snnz 4733	. . . . . . . 8 ⊢ {1o} ≠ ∅
10		dmxp 5910	. . . . . . . 8 ⊢ ({1o} ≠ ∅ → dom (𝑦 × {1o}) = 𝑦)
11	9, 10	ax-mp 5	. . . . . . 7 ⊢ dom (𝑦 × {1o}) = 𝑦
12	11	eqcomi 2769	. . . . . 6 ⊢ 𝑦 = dom (𝑦 × {1o})
13		dmeq 5884	. . . . . . 7 ⊢ (𝑥 = (𝑦 × {1o}) → dom 𝑥 = dom (𝑦 × {1o}))
14	13	rspceeqv 3598	. . . . . 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 28012	. . . . . . 7 ⊢ (𝑥 ∈ No → dom 𝑥 ∈ On)
17		eleq1a 2855	. . . . . . 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 3156	. . . . 5 ⊢ (∃𝑥 ∈ No 𝑦 = dom 𝑥 → 𝑦 ∈ On)
20	15, 19	impbii 212	. . . 4 ⊢ (𝑦 ∈ On ↔ ∃𝑥 ∈ No 𝑦 = dom 𝑥)
21	20	eqabi 2895	. . 3 ⊢ On = {𝑦 ∣ ∃𝑥 ∈ No 𝑦 = dom 𝑥}
22	6, 21	eqtr4i 2786	. 2 ⊢ ran bday = On
23		df-fo 6544	. 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 1569   ∈ wcel 2144  {cab 2738   ≠ wne 2955  ∀wral 3076  ∃wrex 3086  Vcvv 3450  ∅c0 4278  {csn 4584   × cxp 5648  dom cdm 5650  ran crn 5651  Oncon0 6362   Fn wfn 6533  –onto→wfo 6536  1_oc1o 8472  Nocsur 28002  bdaycbday 28004

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-sep 5248  ax-pow 5326  ax-pr 5390  ax-un 7751

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-br 5103  df-opab 5167  df-mpt 5186  df-id 5545  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-suc 6368  df-fun 6540  df-fn 6541  df-f 6542  df-fo 6544  df-1o 8479  df-no 28005  df-bday 28007

This theorem is used by:  nodense  28054  bdayimaon  28055  nosupno  28065  nosupbday  28067  noinfno  28080  noinfbday  28082  noetasuplem4  28098  noetainflem4  28102  bdayfun  28138  bdayfn  28139  bdaydmOLD  28141  bdayrn  28142  bdayon  28143  noprc  28147