Theorem oeiv 6602

Description: Value of ordinal exponentiation. (Contributed by Jim Kingdon, 9-Jul-2019.)

Assertion

Ref	Expression
oeiv	|- ( ( A e. On /\ B e. On ) -> ( A ↑o B ) = ( rec ( ( x e. _V |-> ( x .o A ) ) , 1o ) ` B ) )

Distinct variable group:   x, A

Allowed substitution hint:   B( x)

Proof of Theorem oeiv

Dummy variables y z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		1on 6569	. . 3 |- 1o e. On
2		vex 2802	. . . . . . 7 |- x e. _V
3		omexg 6597	. . . . . . 7 |- ( ( x e. _V /\ A e. On ) -> ( x .o A ) e. _V )
4	2, 3	mpan 424	. . . . . 6 |- ( A e. On -> ( x .o A ) e. _V )
5	4	ralrimivw 2604	. . . . 5 |- ( A e. On -> A. x e. _V ( x .o A ) e. _V )
6		eqid 2229	. . . . . 6 |- ( x e. _V |-> ( x .o A ) ) = ( x e. _V |-> ( x .o A ) )
7	6	fnmpt 5450	. . . . 5 |- ( A. x e. _V ( x .o A ) e. _V -> ( x e. _V |-> ( x .o A ) ) Fn _V )
8	5, 7	syl 14	. . . 4 |- ( A e. On -> ( x e. _V |-> ( x .o A ) ) Fn _V )
9		rdgexggg 6523	. . . 4 |- ( ( ( x e. _V |-> ( x .o A ) ) Fn _V /\ 1o e. On /\ B e. On ) -> ( rec ( ( x e. _V |-> ( x .o A ) ) , 1o ) ` B ) e. _V )
10	8, 9	syl3an1 1304	. . 3 |- ( ( A e. On /\ 1o e. On /\ B e. On ) -> ( rec ( ( x e. _V |-> ( x .o A ) ) , 1o ) ` B ) e. _V )
11	1, 10	mp3an2 1359	. 2 |- ( ( A e. On /\ B e. On ) -> ( rec ( ( x e. _V |-> ( x .o A ) ) , 1o ) ` B ) e. _V )
12		oveq2 6009	. . . . . 6 |- ( y = A -> ( x .o y ) = ( x .o A ) )
13	12	mpteq2dv 4175	. . . . 5 |- ( y = A -> ( x e. _V |-> ( x .o y ) ) = ( x e. _V |-> ( x .o A ) ) )
14		rdgeq1 6517	. . . . 5 |- ( ( x e. _V |-> ( x .o y ) ) = ( x e. _V |-> ( x .o A ) ) -> rec ( ( x e. _V |-> ( x .o y ) ) , 1o ) = rec ( ( x e. _V |-> ( x .o A ) ) , 1o ) )
15	13, 14	syl 14	. . . 4 |- ( y = A -> rec ( ( x e. _V |-> ( x .o y ) ) , 1o ) = rec ( ( x e. _V |-> ( x .o A ) ) , 1o ) )
16	15	fveq1d 5629	. . 3 |- ( y = A -> ( rec ( ( x e. _V |-> ( x .o y ) ) , 1o ) ` z ) = ( rec ( ( x e. _V |-> ( x .o A ) ) , 1o ) ` z ) )
17		fveq2 5627	. . 3 |- ( z = B -> ( rec ( ( x e. _V |-> ( x .o A ) ) , 1o ) ` z ) = ( rec ( ( x e. _V |-> ( x .o A ) ) , 1o ) ` B ) )
18		df-oexpi 6568	. . 3 |- ↑o = ( y e. On , z e. On |-> ( rec ( ( x e. _V |-> ( x .o y ) ) , 1o ) ` z ) )
19	16, 17, 18	ovmpog 6139	. 2 |- ( ( A e. On /\ B e. On /\ ( rec ( ( x e. _V |-> ( x .o A ) ) , 1o ) ` B ) e. _V ) -> ( A ↑o B ) = ( rec ( ( x e. _V |-> ( x .o A ) ) , 1o ) ` B ) )
20	11, 19	mpd3an3 1372	1 |- ( ( A e. On /\ B e. On ) -> ( A ↑o B ) = ( rec ( ( x e. _V |-> ( x .o A ) ) , 1o ) ` B ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1395   e. wcel 2200   A.wral 2508   _Vcvv 2799   |-> cmpt 4145   Oncon0 4454   Fn wfn 5313   ` cfv 5318  (class class class)co 6001   reccrdg 6515   1oc1o 6555   .o comu 6560   ↑_o coei 6561

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4199  ax-sep 4202  ax-nul 4210  ax-pow 4258  ax-pr 4293  ax-un 4524  ax-setind 4629

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-iun 3967  df-br 4084  df-opab 4146  df-mpt 4147  df-tr 4183  df-id 4384  df-iord 4457  df-on 4459  df-suc 4462  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-res 4731  df-ima 4732  df-iota 5278  df-fun 5320  df-fn 5321  df-f 5322  df-f1 5323  df-fo 5324  df-f1o 5325  df-fv 5326  df-ov 6004  df-oprab 6005  df-mpo 6006  df-1st 6286  df-2nd 6287  df-recs 6451  df-irdg 6516  df-1o 6562  df-oadd 6566  df-omul 6567  df-oexpi 6568

This theorem is referenced by:  oei0  6605  oeicl  6608