Theorem bdayfo 27623

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 7909	. . . 4 ⊢ (𝑥 ∈ No → dom 𝑥 ∈ V)
2	1	rgen 3060	. . 3 ⊢ ∀𝑥 ∈ No dom 𝑥 ∈ V
3		df-bday 27591	. . . 4 ⊢ bday = (𝑥 ∈ No ↦ dom 𝑥)
4	3	mptfng 6694	. . 3 ⊢ (∀𝑥 ∈ No dom 𝑥 ∈ V ↔ bday Fn No )
5	2, 4	mpbi 229	. 2 ⊢ bday Fn No
6	3	rnmpt 5957	. . 3 ⊢ ran bday = {𝑦 ∣ ∃𝑥 ∈ No 𝑦 = dom 𝑥}
7		noxp1o 27609	. . . . . 6 ⊢ (𝑦 ∈ On → (𝑦 × {1o}) ∈ No )
8		1oex 8497	. . . . . . . . 9 ⊢ 1o ∈ V
9	8	snnz 4781	. . . . . . . 8 ⊢ {1o} ≠ ∅
10		dmxp 5931	. . . . . . . 8 ⊢ ({1o} ≠ ∅ → dom (𝑦 × {1o}) = 𝑦)
11	9, 10	ax-mp 5	. . . . . . 7 ⊢ dom (𝑦 × {1o}) = 𝑦
12	11	eqcomi 2737	. . . . . 6 ⊢ 𝑦 = dom (𝑦 × {1o})
13		dmeq 5906	. . . . . . 7 ⊢ (𝑥 = (𝑦 × {1o}) → dom 𝑥 = dom (𝑦 × {1o}))
14	13	rspceeqv 3631	. . . . . 6 ⊢ (((𝑦 × {1o}) ∈ No ∧ 𝑦 = dom (𝑦 × {1o})) → ∃𝑥 ∈ No 𝑦 = dom 𝑥)
15	7, 12, 14	sylancl 585	. . . . 5 ⊢ (𝑦 ∈ On → ∃𝑥 ∈ No 𝑦 = dom 𝑥)
16		nodmon 27596	. . . . . . 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 3145	. . . . 5 ⊢ (∃𝑥 ∈ No 𝑦 = dom 𝑥 → 𝑦 ∈ On)
20	15, 19	impbii 208	. . . 4 ⊢ (𝑦 ∈ On ↔ ∃𝑥 ∈ No 𝑦 = dom 𝑥)
21	20	eqabi 2865	. . 3 ⊢ On = {𝑦 ∣ ∃𝑥 ∈ No 𝑦 = dom 𝑥}
22	6, 21	eqtr4i 2759	. 2 ⊢ ran bday = On
23		df-fo 6554	. 2 ⊢ ( bday : No –onto→On ↔ ( bday Fn No ∧ ran bday = On))
24	5, 22, 23	mpbir2an 710	1 ⊢ bday : No –onto→On

Syntax hints:   → wi 4   = wceq 1534   ∈ wcel 2099  {cab 2705   ≠ wne 2937  ∀wral 3058  ∃wrex 3067  Vcvv 3471  ∅c0 4323  {csn 4629   × cxp 5676  dom cdm 5678  ran crn 5679  Oncon0 6369   Fn wfn 6543  –onto→wfo 6546  1_oc1o 8480   No csur 27586   bday cbday 27588

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pr 5429  ax-un 7740

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-reu 3374  df-rab 3430  df-v 3473  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4909  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-suc 6375  df-iota 6500  df-fun 6550  df-fn 6551  df-f 6552  df-f1 6553  df-fo 6554  df-f1o 6555  df-fv 6556  df-1o 8487  df-no 27589  df-bday 27591

This theorem is referenced by:  nodense  27638  bdayimaon  27639  nosupno  27649  nosupbday  27651  noinfno  27664  noinfbday  27666  noetasuplem4  27682  noetainflem4  27686  bdayfun  27718  bdayfn  27719  bdaydm  27720  bdayrn  27721  bdayelon  27722  noprc  27725  noeta2  27730