Theorem bdayfo 27609

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 7826	. . . 4 ⊢ (𝑥 ∈ No → dom 𝑥 ∈ V)
2	1	rgen 3047	. . 3 ⊢ ∀𝑥 ∈ No dom 𝑥 ∈ V
3		df-bday 27576	. . . 4 ⊢ bday = (𝑥 ∈ No ↦ dom 𝑥)
4	3	mptfng 6616	. . 3 ⊢ (∀𝑥 ∈ No dom 𝑥 ∈ V ↔ bday Fn No )
5	2, 4	mpbi 230	. 2 ⊢ bday Fn No
6	3	rnmpt 5894	. . 3 ⊢ ran bday = {𝑦 ∣ ∃𝑥 ∈ No 𝑦 = dom 𝑥}
7		noxp1o 27595	. . . . . 6 ⊢ (𝑦 ∈ On → (𝑦 × {1o}) ∈ No )
8		1oex 8390	. . . . . . . . 9 ⊢ 1o ∈ V
9	8	snnz 4727	. . . . . . . 8 ⊢ {1o} ≠ ∅
10		dmxp 5866	. . . . . . . 8 ⊢ ({1o} ≠ ∅ → dom (𝑦 × {1o}) = 𝑦)
11	9, 10	ax-mp 5	. . . . . . 7 ⊢ dom (𝑦 × {1o}) = 𝑦
12	11	eqcomi 2739	. . . . . 6 ⊢ 𝑦 = dom (𝑦 × {1o})
13		dmeq 5841	. . . . . . 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 586	. . . . 5 ⊢ (𝑦 ∈ On → ∃𝑥 ∈ No 𝑦 = dom 𝑥)
16		nodmon 27582	. . . . . . 7 ⊢ (𝑥 ∈ No → dom 𝑥 ∈ On)
17		eleq1a 2824	. . . . . . 7 ⊢ (dom 𝑥 ∈ On → (𝑦 = dom 𝑥 → 𝑦 ∈ On))
18	16, 17	syl 17	. . . . . 6 ⊢ (𝑥 ∈ No → (𝑦 = dom 𝑥 → 𝑦 ∈ On))
19	18	rexlimiv 3124	. . . . 5 ⊢ (∃𝑥 ∈ No 𝑦 = dom 𝑥 → 𝑦 ∈ On)
20	15, 19	impbii 209	. . . 4 ⊢ (𝑦 ∈ On ↔ ∃𝑥 ∈ No 𝑦 = dom 𝑥)
21	20	eqabi 2864	. . 3 ⊢ On = {𝑦 ∣ ∃𝑥 ∈ No 𝑦 = dom 𝑥}
22	6, 21	eqtr4i 2756	. 2 ⊢ ran bday = On
23		df-fo 6483	. 2 ⊢ ( bday : No –onto→On ↔ ( bday Fn No ∧ ran bday = On))
24	5, 22, 23	mpbir2an 711	1 ⊢ bday : No –onto→On

Syntax hints:   → wi 4   = wceq 1541   ∈ wcel 2110  {cab 2708   ≠ wne 2926  ∀wral 3045  ∃wrex 3054  Vcvv 3434  ∅c0 4281  {csn 4574   × cxp 5612  dom cdm 5614  ran crn 5615  Oncon0 6302   Fn wfn 6472  –onto→wfo 6475  1_oc1o 8373   No csur 27571   bday cbday 27573

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2112  ax-9 2120  ax-10 2143  ax-11 2159  ax-12 2179  ax-ext 2702  ax-sep 5232  ax-nul 5242  ax-pow 5301  ax-pr 5368  ax-un 7663

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-rab 3394  df-v 3436  df-dif 3903  df-un 3905  df-in 3907  df-ss 3917  df-nul 4282  df-if 4474  df-pw 4550  df-sn 4575  df-pr 4577  df-op 4581  df-uni 4858  df-br 5090  df-opab 5152  df-mpt 5171  df-id 5509  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-rn 5625  df-suc 6308  df-fun 6479  df-fn 6480  df-f 6481  df-fo 6483  df-1o 8380  df-no 27574  df-bday 27576

This theorem is referenced by:  nodense  27624  bdayimaon  27625  nosupno  27635  nosupbday  27637  noinfno  27650  noinfbday  27652  noetasuplem4  27668  noetainflem4  27672  bdayfun  27704  bdayfn  27705  bdaydm  27706  bdayrn  27707  bdayelon  27708  noprc  27712  noeta2  27717