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 e. No -> dom x e. _V )
2	1	rgen 2922	. . 3 |- A. x e. No dom x e. _V
3		df-bday 31798	. . . 4 |- bday = ( x e. No |-> dom x )
4	3	mptfng 6019	. . 3 |- ( A. x e. No dom x e. _V <-> bday Fn No )
5	2, 4	mpbi 220	. 2 |- bday Fn No
6	3	rnmpt 5371	. . 3 |- ran bday = { y | E. x e. No y = dom x }
7		noxp1o 31816	. . . . . 6 |- ( y e. On -> ( y X. { 1o } ) e. No )
8		1on 7567	. . . . . . . . . 10 |- 1o e. On
9	8	elexi 3213	. . . . . . . . 9 |- 1o e. _V
10	9	snnz 4309	. . . . . . . 8 |- { 1o } =/= (/)
11		dmxp 5344	. . . . . . . 8 |- ( { 1o } =/= (/) -> dom ( y X. { 1o } ) = y )
12	10, 11	ax-mp 5	. . . . . . 7 |- dom ( y X. { 1o } ) = y
13	12	eqcomi 2631	. . . . . 6 |- y = dom ( y X. { 1o } )
14		dmeq 5324	. . . . . . . 8 |- ( x = ( y X. { 1o } ) -> dom x = dom ( y X. { 1o } ) )
15	14	eqeq2d 2632	. . . . . . 7 |- ( x = ( y X. { 1o } ) -> ( y = dom x <-> y = dom ( y X. { 1o } ) ) )
16	15	rspcev 3309	. . . . . 6 |- ( ( ( y X. { 1o } ) e. No /\ y = dom ( y X. { 1o } ) ) -> E. x e. No y = dom x )
17	7, 13, 16	sylancl 694	. . . . 5 |- ( y e. On -> E. x e. No y = dom x )
18		nodmon 31803	. . . . . . 7 |- ( x e. No -> dom x e. On )
19		eleq1a 2696	. . . . . . 7 |- ( dom x e. On -> ( y = dom x -> y e. On ) )
20	18, 19	syl 17	. . . . . 6 |- ( x e. No -> ( y = dom x -> y e. On ) )
21	20	rexlimiv 3027	. . . . 5 |- ( E. x e. No y = dom x -> y e. On )
22	17, 21	impbii 199	. . . 4 |- ( y e. On <-> E. x e. No y = dom x )
23	22	abbi2i 2738	. . 3 |- On = { y | E. x e. 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   e. wcel 1990   {cab 2608   =/= wne 2794   A.wral 2912   E.wrex 2913   _Vcvv 3200   (/)c0 3915   {csn 4177   X. cxp 5112   dom cdm 5114   ran crn 5115   Oncon0 5723   Fn wfn 5883   -onto->wfo 5886   1oc1o 7553   Nocsur 31793   bdaycbday 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