Theorem bdayimaon 31843

Description: Lemma for full-eta properties. The successor of the union of the image of the birthday function under a set is an ordinal. (Contributed by Scott Fenton, 20-Aug-2011.)

Assertion

Ref	Expression
bdayimaon	|- ( A e. V -> suc U. ( bday " A ) e. On )

Proof of Theorem bdayimaon

Step	Hyp	Ref	Expression
1		bdayfo 31828	. . . . . 6 |- bday : No -onto-> On
2		fofun 6116	. . . . . 6 |- ( bday : No -onto-> On -> Fun bday )
3	1, 2	ax-mp 5	. . . . 5 |- Fun bday
4		funimaexg 5975	. . . . 5 |- ( ( Fun bday /\ A e. V ) -> ( bday " A ) e. _V )
5	3, 4	mpan 706	. . . 4 |- ( A e. V -> ( bday " A ) e. _V )
6		uniexg 6955	. . . 4 |- ( ( bday " A ) e. _V -> U. ( bday " A ) e. _V )
7	5, 6	syl 17	. . 3 |- ( A e. V -> U. ( bday " A ) e. _V )
8		imassrn 5477	. . . . 5 |- ( bday " A ) C_ ran bday
9		forn 6118	. . . . . 6 |- ( bday : No -onto-> On -> ran bday = On )
10	1, 9	ax-mp 5	. . . . 5 |- ran bday = On
11	8, 10	sseqtri 3637	. . . 4 |- ( bday " A ) C_ On
12		ssorduni 6985	. . . 4 |- ( ( bday " A ) C_ On -> Ord U. ( bday " A ) )
13	11, 12	ax-mp 5	. . 3 |- Ord U. ( bday " A )
14	7, 13	jctil 560	. 2 |- ( A e. V -> ( Ord U. ( bday " A ) /\ U. ( bday " A ) e. _V ) )
15		elon2 5734	. . 3 |- ( U. ( bday " A ) e. On <-> ( Ord U. ( bday " A ) /\ U. ( bday " A ) e. _V ) )
16		sucelon 7017	. . 3 |- ( U. ( bday " A ) e. On <-> suc U. ( bday " A ) e. On )
17	15, 16	bitr3i 266	. 2 |- ( ( Ord U. ( bday " A ) /\ U. ( bday " A ) e. _V ) <-> suc U. ( bday " A ) e. On )
18	14, 17	sylib 208	1 |- ( A e. V -> suc U. ( bday " A ) e. On )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   _Vcvv 3200   C_ wss 3574   U.cuni 4436   ran crn 5115   "cima 5117   Ord word 5722   Oncon0 5723   suc csuc 5725   Fun wfun 5882   -onto->wfo 5886   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:  noetalem1  31863