Theorem oemapwe 8017

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 7090	. . . . 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 4840	. . . 4 |- ( ( A ^o B ) e. On -> Ord ( A ^o B ) )
6		ordwe 4843	. . . 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 8016	. . . 4 |- ( ph -> ( A CNF B ) Isom T , _E ( S , ( A ^o B ) ) )
11		isowe 6152	. . . 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 6133	. . . . . 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 6228	. . . . . . . . 9 |- ( A CNF B ) e. _V
18	17	dmex 6624	. . . . . . . 8 |- dom ( A CNF B ) e. _V
19	8, 18	eqeltri 2538	. . . . . . 7 |- S e. _V
20		exse 4795	. . . . . . 7 |- ( S e. _V -> T Se S )
21	19, 20	ax-mp 5	. . . . . 6 |- T Se S
22		eqid 2454	. . . . . . 7 |- OrdIso ( T , S ) = OrdIso ( T , S )
23	22	oieu 7868	. . . . . 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 2462	. 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 2799   E.wrex 2800   _Vcvv 3078   {copab 4460   _E cep 4741   Se wse 4788   We wwe 4789   Ord word 4829   Oncon0 4830   `'ccnv 4950   dom cdm 4951   ` cfv 5529   Isom wiso 5530  (class class class)co 6203   ^o coe 7032  OrdIsocoi 7838   CNF ccnf 7982

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 1955  ax-ext 2432  ax-rep 4514  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642  ax-un 6485

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 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-reu 2806  df-rmo 2807  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3399  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-pss 3455  df-nul 3749  df-if 3903  df-pw 3973  df-sn 3989  df-pr 3991  df-tp 3993  df-op 3995  df-uni 4203  df-int 4240  df-iun 4284  df-br 4404  df-opab 4462  df-mpt 4463  df-tr 4497  df-eprel 4743  df-id 4747  df-po 4752  df-so 4753  df-fr 4790  df-se 4791  df-we 4792  df-ord 4833  df-on 4834  df-lim 4835  df-suc 4836  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-res 4963  df-ima 4964  df-iota 5492  df-fun 5531  df-fn 5532  df-f 5533  df-f1 5534  df-fo 5535  df-f1o 5536  df-fv 5537  df-isom 5538  df-riota 6164  df-ov 6206  df-oprab 6207  df-mpt2 6208  df-om 6590  df-1st 6690  df-2nd 6691  df-supp 6804  df-recs 6945  df-rdg 6979  df-seqom 7016  df-1o 7033  df-2o 7034  df-oadd 7037  df-omul 7038  df-oexp 7039  df-er 7214  df-map 7329  df-en 7424  df-dom 7425  df-sdom 7426  df-fin 7427  df-fsupp 7735  df-oi 7839  df-cnf 7983

This theorem is referenced by:  cantnffval2  8018  cantnffval2OLD  8040  wemapwe  8043  wemapweOLD  8044