Theorem oemapwe 8144

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.)

Hypotheses

Ref	Expression
cantnfs.s	|- S = dom ( A CNF B )
cantnfs.a	|- ( ph -> A e. On )
cantnfs.b	|- ( ph -> B e. On )
oemapval.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
oemapwe	|- ( 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 oemapwe

Step	Hyp	Ref	Expression
1		cantnfs.a	. . . . 5 |- ( ph -> A e. On )
2		cantnfs.b	. . . . 5 |- ( ph -> B e. On )
3		oecl 7223	. . . . 5 |- ( ( A e. On /\ B e. On ) -> ( A ^o B ) e. On )
4	1, 2, 3	syl2anc 659	. . . 4 |- ( ph -> ( A ^o B ) e. On )
5		eloni 5419	. . . 4 |- ( ( A ^o B ) e. On -> Ord ( A ^o B ) )
6		ordwe 5422	. . . 4 |- ( Ord ( A ^o B ) -> _E We ( A ^o B ) )
7	4, 5, 6	3syl 20	. . 3 |- ( ph -> _E We ( A ^o B ) )
8		cantnfs.s	. . . . 5 |- S = dom ( A CNF B )
9		oemapval.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	cantnf 8143	. . . 4 |- ( ph -> ( A CNF B ) Isom T , _E ( S , ( A ^o B ) ) )
11		isowe 6227	. . . 4 |- ( ( A CNF B ) Isom T , _E ( S , ( A ^o B ) ) -> ( T We S <-> _E We ( A ^o B ) ) )
12	10, 11	syl 17	. . 3 |- ( ph -> ( T We S <-> _E We ( A ^o B ) ) )
13	7, 12	mpbird 232	. 2 |- ( ph -> T We S )
14	4, 5	syl 17	. . . . 5 |- ( ph -> Ord ( A ^o B ) )
15		isocnv 6208	. . . . . 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 17	. . . . 5 |- ( ph -> `' ( A CNF B ) Isom _E , T ( ( A ^o B ) , S ) )
17		ovex 6305	. . . . . . . . 9 |- ( A CNF B ) e. _V
18	17	dmex 6716	. . . . . . . 8 |- dom ( A CNF B ) e. _V
19	8, 18	eqeltri 2486	. . . . . . 7 |- S e. _V
20		exse 4786	. . . . . . 7 |- ( S e. _V -> T Se S )
21	19, 20	ax-mp 5	. . . . . 6 |- T Se S
22		eqid 2402	. . . . . . 7 |- OrdIso ( T , S ) = OrdIso ( T , S )
23	22	oieu 7997	. . . . . 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 660	. . . . 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 922	. . . 4 |- ( ph -> ( ( A ^o B ) = dom OrdIso ( T , S ) /\ `' ( A CNF B ) = OrdIso ( T , S ) ) )
26	25	simpld 457	. . 3 |- ( ph -> ( A ^o B ) = dom OrdIso ( T , S ) )
27	26	eqcomd 2410	. 2 |- ( ph -> dom OrdIso ( T , S ) = ( A ^o B ) )
28	13, 27	jca 530	1 |- ( ph -> ( T We S /\ dom OrdIso ( T , S ) = ( A ^o B ) ) )

Syntax hints:   -> wi 4   <-> wb 184   /\ wa 367   = wceq 1405   e. wcel 1842   A.wral 2753   E.wrex 2754   _Vcvv 3058   {copab 4451   _E cep 4731   Se wse 4779   We wwe 4780   `'ccnv 4821   dom cdm 4822   Ord word 5408   Oncon0 5409   ` cfv 5568   Isom wiso 5569  (class class class)co 6277   ^o coe 7165  OrdIsocoi 7967   CNF ccnf 8109

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4506  ax-sep 4516  ax-nul 4524  ax-pow 4571  ax-pr 4629  ax-un 6573

This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 975  df-3an 976  df-tru 1408  df-fal 1411  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2758  df-rex 2759  df-reu 2760  df-rmo 2761  df-rab 2762  df-v 3060  df-sbc 3277  df-csb 3373  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-pss 3429  df-nul 3738  df-if 3885  df-pw 3956  df-sn 3972  df-pr 3974  df-tp 3976  df-op 3978  df-uni 4191  df-int 4227  df-iun 4272  df-br 4395  df-opab 4453  df-mpt 4454  df-tr 4489  df-eprel 4733  df-id 4737  df-po 4743  df-so 4744  df-fr 4781  df-se 4782  df-we 4783  df-xp 4828  df-rel 4829  df-cnv 4830  df-co 4831  df-dm 4832  df-rn 4833  df-res 4834  df-ima 4835  df-pred 5366  df-ord 5412  df-on 5413  df-lim 5414  df-suc 5415  df-iota 5532  df-fun 5570  df-fn 5571  df-f 5572  df-f1 5573  df-fo 5574  df-f1o 5575  df-fv 5576  df-isom 5577  df-riota 6239  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-om 6683  df-1st 6783  df-2nd 6784  df-supp 6902  df-wrecs 7012  df-recs 7074  df-rdg 7112  df-seqom 7149  df-1o 7166  df-2o 7167  df-oadd 7170  df-omul 7171  df-oexp 7172  df-er 7347  df-map 7458  df-en 7554  df-dom 7555  df-sdom 7556  df-fin 7557  df-fsupp 7863  df-oi 7968  df-cnf 8110

This theorem is referenced by:  cantnffval2  8145  cantnffval2OLD  8167  wemapwe  8170  wemapweOLD  8171