Theorem oemapweOLD 8022

Description: The lexicographic order on a function space of ordinals gives a well-ordering with order type equal to the ordinal exponential. This provides an alternative definition of the ordinal exponential. (Contributed by Mario Carneiro, 28-May-2015.) Obsolete version of oemapwe 8000 as of 2-Jul-2019. (New usage is discouraged.)

Hypotheses

Ref	Expression
cantnfsOLD.1	|- S = dom ( A CNF B )
cantnfsOLD.2	|- ( ph -> A e. On )
cantnfsOLD.3	|- ( ph -> B e. On )
oemapvalOLD.t	|- T = { <. x , y >. | E. z e. B ( ( x ` z ) e. ( y ` z ) /\ A. w e. B ( z e. w -> ( x ` w ) = ( y ` w ) ) ) }

Assertion

Ref	Expression
oemapweOLD	|- ( ph -> ( T We S /\ dom OrdIso ( T , S ) = ( A ^o B ) ) )

Distinct variable groups:   x, w, y, z, B   w, A, x, y, z   x, S, y, z   ph, x, y, z

Allowed substitution hints:   ph( w)   S( w)   T( x, y, z, w)

Proof of Theorem oemapweOLD

Step	Hyp	Ref	Expression
1		cantnfsOLD.2	. . . . 5 |- ( ph -> A e. On )
2		cantnfsOLD.3	. . . . 5 |- ( ph -> B e. On )
3		oecl 7074	. . . . 5 |- ( ( A e. On /\ B e. On ) -> ( A ^o B ) e. On )
4	1, 2, 3	syl2anc 661	. . . 4 |- ( ph -> ( A ^o B ) e. On )
5		eloni 4824	. . . 4 |- ( ( A ^o B ) e. On -> Ord ( A ^o B ) )
6		ordwe 4827	. . . 4 |- ( Ord ( A ^o B ) -> _E We ( A ^o B ) )
7	4, 5, 6	3syl 20	. . 3 |- ( ph -> _E We ( A ^o B ) )
8		cantnfsOLD.1	. . . . 5 |- S = dom ( A CNF B )
9		oemapvalOLD.t	. . . . 5 |- T = { <. x , y >. | E. z e. B ( ( x ` z ) e. ( y ` z ) /\ A. w e. B ( z e. w -> ( x ` w ) = ( y ` w ) ) ) }
10	8, 1, 2, 9	cantnfOLD 8021	. . . 4 |- ( ph -> ( A CNF B ) Isom T , _E ( S , ( A ^o B ) ) )
11		isowe 6136	. . . 4 |- ( ( A CNF B ) Isom T , _E ( S , ( A ^o B ) ) -> ( T We S <-> _E We ( A ^o B ) ) )
12	10, 11	syl 16	. . 3 |- ( ph -> ( T We S <-> _E We ( A ^o B ) ) )
13	7, 12	mpbird 232	. 2 |- ( ph -> T We S )
14	4, 5	syl 16	. . . . 5 |- ( ph -> Ord ( A ^o B ) )
15		isocnv 6117	. . . . . 6 |- ( ( A CNF B ) Isom T , _E ( S , ( A ^o B ) ) -> `' ( A CNF B ) Isom _E , T ( ( A ^o B ) , S ) )
16	10, 15	syl 16	. . . . 5 |- ( ph -> `' ( A CNF B ) Isom _E , T ( ( A ^o B ) , S ) )
17		ovex 6212	. . . . . . . . 9 |- ( A CNF B ) e. _V
18	17	dmex 6608	. . . . . . . 8 |- dom ( A CNF B ) e. _V
19	8, 18	eqeltri 2533	. . . . . . 7 |- S e. _V
20		exse 4779	. . . . . . 7 |- ( S e. _V -> T Se S )
21	19, 20	ax-mp 5	. . . . . 6 |- T Se S
22		eqid 2451	. . . . . . 7 |- OrdIso ( T , S ) = OrdIso ( T , S )
23	22	oieu 7851	. . . . . 6 |- ( ( T We S /\ T Se S ) -> ( ( Ord ( A ^o B ) /\ `' ( A CNF B ) Isom _E , T ( ( A ^o B ) , S ) ) <-> ( ( A ^o B ) = dom OrdIso ( T , S ) /\ `' ( A CNF B ) = OrdIso ( T , S ) ) ) )
24	13, 21, 23	sylancl 662	. . . . 5 |- ( ph -> ( ( Ord ( A ^o B ) /\ `' ( A CNF B ) Isom _E , T ( ( A ^o B ) , S ) ) <-> ( ( A ^o B ) = dom OrdIso ( T , S ) /\ `' ( A CNF B ) = OrdIso ( T , S ) ) ) )
25	14, 16, 24	mpbi2and 912	. . . 4 |- ( ph -> ( ( A ^o B ) = dom OrdIso ( T , S ) /\ `' ( A CNF B ) = OrdIso ( T , S ) ) )
26	25	simpld 459	. . 3 |- ( ph -> ( A ^o B ) = dom OrdIso ( T , S ) )
27	26	eqcomd 2458	. 2 |- ( ph -> dom OrdIso ( T , S ) = ( A ^o B ) )
28	13, 27	jca 532	1 |- ( ph -> ( T We S /\ dom OrdIso ( T , S ) = ( A ^o B ) ) )

Syntax hints:   -> wi 4   <-> wb 184   /\ wa 369   = wceq 1370   e. wcel 1758   A.wral 2793   E.wrex 2794   _Vcvv 3065   {copab 4444   _E cep 4725   Se wse 4772   We wwe 4773   Ord word 4813   Oncon0 4814   `'ccnv 4934   dom cdm 4935   ` cfv 5513   Isom wiso 5514  (class class class)co 6187   ^o coe 7016  OrdIsocoi 7821   CNF ccnf 7965

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-rep 4498  ax-sep 4508  ax-nul 4516  ax-pow 4565  ax-pr 4626  ax-un 6469

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-fal 1376  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2599  df-ne 2644  df-ral 2798  df-rex 2799  df-reu 2800  df-rmo 2801  df-rab 2802  df-v 3067  df-sbc 3282  df-csb 3384  df-dif 3426  df-un 3428  df-in 3430  df-ss 3437  df-pss 3439  df-nul 3733  df-if 3887  df-pw 3957  df-sn 3973  df-pr 3975  df-tp 3977  df-op 3979  df-uni 4187  df-int 4224  df-iun 4268  df-br 4388  df-opab 4446  df-mpt 4447  df-tr 4481  df-eprel 4727  df-id 4731  df-po 4736  df-so 4737  df-fr 4774  df-se 4775  df-we 4776  df-ord 4817  df-on 4818  df-lim 4819  df-suc 4820  df-xp 4941  df-rel 4942  df-cnv 4943  df-co 4944  df-dm 4945  df-rn 4946  df-res 4947  df-ima 4948  df-iota 5476  df-fun 5515  df-fn 5516  df-f 5517  df-f1 5518  df-fo 5519  df-f1o 5520  df-fv 5521  df-isom 5522  df-riota 6148  df-ov 6190  df-oprab 6191  df-mpt2 6192  df-om 6574  df-1st 6674  df-2nd 6675  df-supp 6788  df-recs 6929  df-rdg 6963  df-seqom 7000  df-1o 7017  df-2o 7018  df-oadd 7021  df-omul 7022  df-oexp 7023  df-er 7198  df-map 7313  df-en 7408  df-dom 7409  df-sdom 7410  df-fin 7411  df-fsupp 7719  df-oi 7822  df-cnf 7966

This theorem is referenced by: (None)