Theorem oemapwe 7606

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.1	|- S = dom ( A CNF B )
cantnfs.2	|- ( ph -> A e. On )
cantnfs.3	|- ( 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.2	. . . . 5 |- ( ph -> A e. On )
2		cantnfs.3	. . . . 5 |- ( ph -> B e. On )
3		oecl 6740	. . . . 5 |- ( ( A e. On /\ B e. On ) -> ( A ^o B ) e. On )
4	1, 2, 3	syl2anc 643	. . . 4 |- ( ph -> ( A ^o B ) e. On )
5		eloni 4551	. . . 4 |- ( ( A ^o B ) e. On -> Ord ( A ^o B ) )
6		ordwe 4554	. . . 4 |- ( Ord ( A ^o B ) -> _E We ( A ^o B ) )
7	4, 5, 6	3syl 19	. . 3 |- ( ph -> _E We ( A ^o B ) )
8		cantnfs.1	. . . . 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 7605	. . . 4 |- ( ph -> ( A CNF B ) Isom T , _E ( S , ( A ^o B ) ) )
11		isowe 6028	. . . 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 224	. 2 |- ( ph -> T We S )
14	4, 5	syl 16	. . . . 5 |- ( ph -> Ord ( A ^o B ) )
15		isocnv 6009	. . . . . 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 6065	. . . . . . . . 9 |- ( A CNF B ) e. _V
18	17	dmex 5091	. . . . . . . 8 |- dom ( A CNF B ) e. _V
19	8, 18	eqeltri 2474	. . . . . . 7 |- S e. _V
20		exse 4506	. . . . . . 7 |- ( S e. _V -> T Se S )
21	19, 20	ax-mp 8	. . . . . 6 |- T Se S
22		eqid 2404	. . . . . . 7 |- OrdIso ( T , S ) = OrdIso ( T , S )
23	22	oieu 7464	. . . . . 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 644	. . . . 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 888	. . . 4 |- ( ph -> ( ( A ^o B ) = dom OrdIso ( T , S ) /\ `' ( A CNF B ) = OrdIso ( T , S ) ) )
26	25	simpld 446	. . 3 |- ( ph -> ( A ^o B ) = dom OrdIso ( T , S ) )
27	26	eqcomd 2409	. 2 |- ( ph -> dom OrdIso ( T , S ) = ( A ^o B ) )
28	13, 27	jca 519	1 |- ( ph -> ( T We S /\ dom OrdIso ( T , S ) = ( A ^o B ) ) )

Syntax hints:   -> wi 4   <-> wb 177   /\ wa 359   = wceq 1649   e. wcel 1721   A.wral 2666   E.wrex 2667   _Vcvv 2916   {copab 4225   _E cep 4452   Se wse 4499   We wwe 4500   Ord word 4540   Oncon0 4541   `'ccnv 4836   dom cdm 4837   ` cfv 5413   Isom wiso 5414  (class class class)co 6040   ^o coe 6682  OrdIsocoi 7434   CNF ccnf 7572

This theorem is referenced by:  cantnffval2  7607  wemapwe  7610

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660

This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-int 4011  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-se 4502  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-isom 5422  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-1st 6308  df-2nd 6309  df-riota 6508  df-recs 6592  df-rdg 6627  df-seqom 6664  df-1o 6683  df-2o 6684  df-oadd 6687  df-omul 6688  df-oexp 6689  df-er 6864  df-map 6979  df-en 7069  df-dom 7070  df-sdom 7071  df-fin 7072  df-oi 7435  df-cnf 7573