Theorem bdayfo 31074

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 6989	. . . 4 ⊢ (𝑥 ∈ No → dom 𝑥 ∈ V)
2	1	rgen 2906	. . 3 ⊢ ∀𝑥 ∈ No dom 𝑥 ∈ V
3		df-bday 31042	. . . 4 ⊢ bday = (𝑥 ∈ No ↦ dom 𝑥)
4	3	mptfng 5932	. . 3 ⊢ (∀𝑥 ∈ No dom 𝑥 ∈ V ↔ bday Fn No )
5	2, 4	mpbi 219	. 2 ⊢ bday Fn No
6	3	rnmpt 5292	. . 3 ⊢ ran bday = {𝑦 ∣ ∃𝑥 ∈ No 𝑦 = dom 𝑥}
7		noxp1o 31063	. . . . . 6 ⊢ (𝑦 ∈ On → (𝑦 × {1𝑜}) ∈ No )
8		1on 7454	. . . . . . . . . 10 ⊢ 1𝑜 ∈ On
9	8	elexi 3186	. . . . . . . . 9 ⊢ 1𝑜 ∈ V
10	9	snnz 4252	. . . . . . . 8 ⊢ {1𝑜} ≠ ∅
11		dmxp 5265	. . . . . . . 8 ⊢ ({1𝑜} ≠ ∅ → dom (𝑦 × {1𝑜}) = 𝑦)
12	10, 11	ax-mp 5	. . . . . . 7 ⊢ dom (𝑦 × {1𝑜}) = 𝑦
13	12	eqcomi 2619	. . . . . 6 ⊢ 𝑦 = dom (𝑦 × {1𝑜})
14		dmeq 5246	. . . . . . . 8 ⊢ (𝑥 = (𝑦 × {1𝑜}) → dom 𝑥 = dom (𝑦 × {1𝑜}))
15	14	eqeq2d 2620	. . . . . . 7 ⊢ (𝑥 = (𝑦 × {1𝑜}) → (𝑦 = dom 𝑥 ↔ 𝑦 = dom (𝑦 × {1𝑜})))
16	15	rspcev 3282	. . . . . 6 ⊢ (((𝑦 × {1𝑜}) ∈ No ∧ 𝑦 = dom (𝑦 × {1𝑜})) → ∃𝑥 ∈ No 𝑦 = dom 𝑥)
17	7, 13, 16	sylancl 693	. . . . 5 ⊢ (𝑦 ∈ On → ∃𝑥 ∈ No 𝑦 = dom 𝑥)
18		nodmon 31047	. . . . . . 7 ⊢ (𝑥 ∈ No → dom 𝑥 ∈ On)
19		eleq1a 2683	. . . . . . 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 3009	. . . . 5 ⊢ (∃𝑥 ∈ No 𝑦 = dom 𝑥 → 𝑦 ∈ On)
22	17, 21	impbii 198	. . . 4 ⊢ (𝑦 ∈ On ↔ ∃𝑥 ∈ No 𝑦 = dom 𝑥)
23	22	abbi2i 2725	. . 3 ⊢ On = {𝑦 ∣ ∃𝑥 ∈ No 𝑦 = dom 𝑥}
24	6, 23	eqtr4i 2635	. 2 ⊢ ran bday = On
25		df-fo 5810	. 2 ⊢ ( bday : No –onto→On ↔ ( bday Fn No ∧ ran bday = On))
26	5, 24, 25	mpbir2an 957	1 ⊢ bday : No –onto→On

Syntax hints:   → wi 4   = wceq 1475   ∈ wcel 1977  {cab 2596   ≠ wne 2780  ∀wral 2896  ∃wrex 2897  Vcvv 3173  ∅c0 3874  {csn 4125   × cxp 5036  dom cdm 5038  ran crn 5039  Oncon0 5640   Fn wfn 5799  –onto→wfo 5802  1_𝑜c1o 7440   No csur 31037   bday cbday 31039

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pr 4833  ax-un 6847

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-ord 5643  df-on 5644  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-1o 7447  df-no 31040  df-bday 31042

This theorem is referenced by:  bdayfun  31075  bdayrn  31076  bdaydm  31077  noprc  31080