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Theorem oev 6513
Description: Value of ordinal exponentiation. (Contributed by NM, 30-Dec-2004.) (Revised by Mario Carneiro, 23-Aug-2014.)
Assertion
Ref Expression
oev  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( A  ^o  B
)  =  if ( A  =  (/) ,  ( 1o  \  B ) ,  ( rec (
( x  e.  _V  |->  ( x  .o  A
) ) ,  1o ) `  B )
) )
Distinct variable group:    x, A
Allowed substitution hint:    B( x)

Proof of Theorem oev
Dummy variables  y 
z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqeq1 2289 . . 3  |-  ( y  =  A  ->  (
y  =  (/)  <->  A  =  (/) ) )
2 oveq2 5866 . . . . . 6  |-  ( y  =  A  ->  (
x  .o  y )  =  ( x  .o  A ) )
32mpteq2dv 4107 . . . . 5  |-  ( y  =  A  ->  (
x  e.  _V  |->  ( x  .o  y ) )  =  ( x  e.  _V  |->  ( x  .o  A ) ) )
4 rdgeq1 6424 . . . . 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 ) )
53, 4syl 15 . . . 4  |-  ( y  =  A  ->  rec ( ( x  e. 
_V  |->  ( x  .o  y ) ) ,  1o )  =  rec ( ( x  e. 
_V  |->  ( x  .o  A ) ) ,  1o ) )
65fveq1d 5527 . . 3  |-  ( y  =  A  ->  ( rec ( ( x  e. 
_V  |->  ( x  .o  y ) ) ,  1o ) `  z
)  =  ( rec ( ( x  e. 
_V  |->  ( x  .o  A ) ) ,  1o ) `  z
) )
71, 6ifbieq2d 3585 . 2  |-  ( y  =  A  ->  if ( y  =  (/) ,  ( 1o  \  z
) ,  ( rec ( ( x  e. 
_V  |->  ( x  .o  y ) ) ,  1o ) `  z
) )  =  if ( A  =  (/) ,  ( 1o  \  z
) ,  ( rec ( ( x  e. 
_V  |->  ( x  .o  A ) ) ,  1o ) `  z
) ) )
8 difeq2 3288 . . 3  |-  ( z  =  B  ->  ( 1o  \  z )  =  ( 1o  \  B
) )
9 fveq2 5525 . . 3  |-  ( z  =  B  ->  ( rec ( ( x  e. 
_V  |->  ( x  .o  A ) ) ,  1o ) `  z
)  =  ( rec ( ( x  e. 
_V  |->  ( x  .o  A ) ) ,  1o ) `  B
) )
108, 9ifeq12d 3581 . 2  |-  ( z  =  B  ->  if ( A  =  (/) ,  ( 1o  \  z ) ,  ( rec (
( x  e.  _V  |->  ( x  .o  A
) ) ,  1o ) `  z )
)  =  if ( A  =  (/) ,  ( 1o  \  B ) ,  ( rec (
( x  e.  _V  |->  ( x  .o  A
) ) ,  1o ) `  B )
) )
11 df-oexp 6485 . 2  |-  ^o  =  ( y  e.  On ,  z  e.  On  |->  if ( y  =  (/) ,  ( 1o  \  z
) ,  ( rec ( ( x  e. 
_V  |->  ( x  .o  y ) ) ,  1o ) `  z
) ) )
12 1on 6486 . . . . 5  |-  1o  e.  On
1312elexi 2797 . . . 4  |-  1o  e.  _V
14 difss 3303 . . . 4  |-  ( 1o 
\  B )  C_  1o
1513, 14ssexi 4159 . . 3  |-  ( 1o 
\  B )  e. 
_V
16 fvex 5539 . . 3  |-  ( rec ( ( x  e. 
_V  |->  ( x  .o  A ) ) ,  1o ) `  B
)  e.  _V
1715, 16ifex 3623 . 2  |-  if ( A  =  (/) ,  ( 1o  \  B ) ,  ( rec (
( x  e.  _V  |->  ( x  .o  A
) ) ,  1o ) `  B )
)  e.  _V
187, 10, 11, 17ovmpt2 5983 1  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( A  ^o  B
)  =  if ( A  =  (/) ,  ( 1o  \  B ) ,  ( rec (
( x  e.  _V  |->  ( x  .o  A
) ) ,  1o ) `  B )
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684   _Vcvv 2788    \ cdif 3149   (/)c0 3455   ifcif 3565    e. cmpt 4077   Oncon0 4392   ` cfv 5255  (class class class)co 5858   reccrdg 6422   1oc1o 6472    .o comu 6477    ^o coe 6478
This theorem is referenced by:  oevn0  6514  oe0m  6517
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-suc 4398  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-iota 5219  df-fun 5257  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-recs 6388  df-rdg 6423  df-1o 6479  df-oexp 6485
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