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Theorem oev 7176
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 2471 . . 3  |-  ( y  =  A  ->  (
y  =  (/)  <->  A  =  (/) ) )
2 oveq2 6303 . . . . . 6  |-  ( y  =  A  ->  (
x  .o  y )  =  ( x  .o  A ) )
32mpteq2dv 4540 . . . . 5  |-  ( y  =  A  ->  (
x  e.  _V  |->  ( x  .o  y ) )  =  ( x  e.  _V  |->  ( x  .o  A ) ) )
4 rdgeq1 7089 . . . . 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 16 . . . 4  |-  ( y  =  A  ->  rec ( ( x  e. 
_V  |->  ( x  .o  y ) ) ,  1o )  =  rec ( ( x  e. 
_V  |->  ( x  .o  A ) ) ,  1o ) )
65fveq1d 5874 . . 3  |-  ( y  =  A  ->  ( rec ( ( x  e. 
_V  |->  ( x  .o  y ) ) ,  1o ) `  z
)  =  ( rec ( ( x  e. 
_V  |->  ( x  .o  A ) ) ,  1o ) `  z
) )
71, 6ifbieq2d 3970 . 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 3621 . . 3  |-  ( z  =  B  ->  ( 1o  \  z )  =  ( 1o  \  B
) )
9 fveq2 5872 . . 3  |-  ( z  =  B  ->  ( rec ( ( x  e. 
_V  |->  ( x  .o  A ) ) ,  1o ) `  z
)  =  ( rec ( ( x  e. 
_V  |->  ( x  .o  A ) ) ,  1o ) `  B
) )
108, 9ifeq12d 3965 . 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 7148 . 2  |-  ^o  =  ( y  e.  On ,  z  e.  On  |->  if ( y  =  (/) ,  ( 1o  \  z
) ,  ( rec ( ( x  e. 
_V  |->  ( x  .o  y ) ) ,  1o ) `  z
) ) )
12 1on 7149 . . . . 5  |-  1o  e.  On
1312elexi 3128 . . . 4  |-  1o  e.  _V
14 difss 3636 . . . 4  |-  ( 1o 
\  B )  C_  1o
1513, 14ssexi 4598 . . 3  |-  ( 1o 
\  B )  e. 
_V
16 fvex 5882 . . 3  |-  ( rec ( ( x  e. 
_V  |->  ( x  .o  A ) ) ,  1o ) `  B
)  e.  _V
1715, 16ifex 4014 . 2  |-  if ( A  =  (/) ,  ( 1o  \  B ) ,  ( rec (
( x  e.  _V  |->  ( x  .o  A
) ) ,  1o ) `  B )
)  e.  _V
187, 10, 11, 17ovmpt2 6433 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 setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1379    e. wcel 1767   _Vcvv 3118    \ cdif 3478   (/)c0 3790   ifcif 3945    |-> cmpt 4511   Oncon0 4884   ` cfv 5594  (class class class)co 6295   reccrdg 7087   1oc1o 7135    .o comu 7140    ^o coe 7141
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4574  ax-nul 4582  ax-pr 4692  ax-un 6587
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-rab 2826  df-v 3120  df-sbc 3337  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-tp 4038  df-op 4040  df-uni 4252  df-br 4454  df-opab 4512  df-mpt 4513  df-tr 4547  df-eprel 4797  df-id 4801  df-po 4806  df-so 4807  df-fr 4844  df-we 4846  df-ord 4887  df-on 4888  df-suc 4890  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-iota 5557  df-fun 5596  df-fv 5602  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-recs 7054  df-rdg 7088  df-1o 7142  df-oexp 7148
This theorem is referenced by:  oevn0  7177  oe0m  7180
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