Theorem bdayfo 31828

Description: The birthday function maps the surreals onto the ordinals. Alling's axiom (B). (Shortened proof on 2012-Apr-14, 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 7097	. . . 4 ⊢ (𝑥 ∈ No → dom 𝑥 ∈ V)
2	1	rgen 2922	. . 3 ⊢ ∀𝑥 ∈ No dom 𝑥 ∈ V
3		df-bday 31798	. . . 4 ⊢ bday = (𝑥 ∈ No ↦ dom 𝑥)
4	3	mptfng 6019	. . 3 ⊢ (∀𝑥 ∈ No dom 𝑥 ∈ V ↔ bday Fn No )
5	2, 4	mpbi 220	. 2 ⊢ bday Fn No
6	3	rnmpt 5371	. . 3 ⊢ ran bday = {𝑦 ∣ ∃𝑥 ∈ No 𝑦 = dom 𝑥}
7		noxp1o 31816	. . . . . 6 ⊢ (𝑦 ∈ On → (𝑦 × {1𝑜}) ∈ No )
8		1on 7567	. . . . . . . . . 10 ⊢ 1𝑜 ∈ On
9	8	elexi 3213	. . . . . . . . 9 ⊢ 1𝑜 ∈ V
10	9	snnz 4309	. . . . . . . 8 ⊢ {1𝑜} ≠ ∅
11		dmxp 5344	. . . . . . . 8 ⊢ ({1𝑜} ≠ ∅ → dom (𝑦 × {1𝑜}) = 𝑦)
12	10, 11	ax-mp 5	. . . . . . 7 ⊢ dom (𝑦 × {1𝑜}) = 𝑦
13	12	eqcomi 2631	. . . . . 6 ⊢ 𝑦 = dom (𝑦 × {1𝑜})
14		dmeq 5324	. . . . . . . 8 ⊢ (𝑥 = (𝑦 × {1𝑜}) → dom 𝑥 = dom (𝑦 × {1𝑜}))
15	14	eqeq2d 2632	. . . . . . 7 ⊢ (𝑥 = (𝑦 × {1𝑜}) → (𝑦 = dom 𝑥 ↔ 𝑦 = dom (𝑦 × {1𝑜})))
16	15	rspcev 3309	. . . . . 6 ⊢ (((𝑦 × {1𝑜}) ∈ No ∧ 𝑦 = dom (𝑦 × {1𝑜})) → ∃𝑥 ∈ No 𝑦 = dom 𝑥)
17	7, 13, 16	sylancl 694	. . . . 5 ⊢ (𝑦 ∈ On → ∃𝑥 ∈ No 𝑦 = dom 𝑥)
18		nodmon 31803	. . . . . . 7 ⊢ (𝑥 ∈ No → dom 𝑥 ∈ On)
19		eleq1a 2696	. . . . . . 7 ⊢ (dom 𝑥 ∈ On → (𝑦 = dom 𝑥 → 𝑦 ∈ On))
20	18, 19	syl 17	. . . . . 6 ⊢ (𝑥 ∈ No → (𝑦 = dom 𝑥 → 𝑦 ∈ On))
21	20	rexlimiv 3027	. . . . 5 ⊢ (∃𝑥 ∈ No 𝑦 = dom 𝑥 → 𝑦 ∈ On)
22	17, 21	impbii 199	. . . 4 ⊢ (𝑦 ∈ On ↔ ∃𝑥 ∈ No 𝑦 = dom 𝑥)
23	22	abbi2i 2738	. . 3 ⊢ On = {𝑦 ∣ ∃𝑥 ∈ No 𝑦 = dom 𝑥}
24	6, 23	eqtr4i 2647	. 2 ⊢ ran bday = On
25		df-fo 5894	. 2 ⊢ ( bday : No –onto→On ↔ ( bday Fn No ∧ ran bday = On))
26	5, 24, 25	mpbir2an 955	1 ⊢ bday : No –onto→On

Syntax hints:   → wi 4   = wceq 1483   ∈ wcel 1990  {cab 2608   ≠ wne 2794  ∀wral 2912  ∃wrex 2913  Vcvv 3200  ∅c0 3915  {csn 4177   × cxp 5112  dom cdm 5114  ran crn 5115  Oncon0 5723   Fn wfn 5883  –onto→wfo 5886  1_𝑜c1o 7553   No csur 31793   bday cbday 31795

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-ord 5726  df-on 5727  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-1o 7560  df-no 31796  df-bday 31798

This theorem is referenced by:  nodense  31842  bdayimaon  31843  nosupno  31849  nosupbday  31851  noetalem3  31865  noetalem4  31866  bdayfun  31888  bdayfn  31889  bdaydm  31890  bdayrn  31891  bdayelon  31892  noprc  31895