Theorem oemapweOLD 8126

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 8104 as of 2-Jul-2019. (New usage is discouraged.) (Proof modification 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 7179	. . . . 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 4877	. . . 4 |- ( ( A ^o B ) e. On -> Ord ( A ^o B ) )
6		ordwe 4880	. . . 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 8125	. . . 4 |- ( ph -> ( A CNF B ) Isom T , _E ( S , ( A ^o B ) ) )
11		isowe 6220	. . . 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 6201	. . . . . 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 6298	. . . . . . . . 9 |- ( A CNF B ) e. _V
18	17	dmex 6706	. . . . . . . 8 |- dom ( A CNF B ) e. _V
19	8, 18	eqeltri 2538	. . . . . . 7 |- S e. _V
20		exse 4832	. . . . . . 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 7956	. . . . . 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 919	. . . 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 2462	. 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 1398   e. wcel 1823   A.wral 2804   E.wrex 2805   _Vcvv 3106   {copab 4496   _E cep 4778   Se wse 4825   We wwe 4826   Ord word 4866   Oncon0 4867   `'ccnv 4987   dom cdm 4988   ` cfv 5570   Isom wiso 5571  (class class class)co 6270   ^o coe 7121  OrdIsocoi 7926   CNF ccnf 8069

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565

This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-fal 1404  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-reu 2811  df-rmo 2812  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-int 4272  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-se 4828  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-isom 5579  df-riota 6232  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-om 6674  df-1st 6773  df-2nd 6774  df-supp 6892  df-recs 7034  df-rdg 7068  df-seqom 7105  df-1o 7122  df-2o 7123  df-oadd 7126  df-omul 7127  df-oexp 7128  df-er 7303  df-map 7414  df-en 7510  df-dom 7511  df-sdom 7512  df-fin 7513  df-fsupp 7822  df-oi 7927  df-cnf 8070

This theorem is referenced by: (None)