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Theorem oev 6597
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 2364 . . 3  |-  ( y  =  A  ->  (
y  =  (/)  <->  A  =  (/) ) )
2 oveq2 5950 . . . . . 6  |-  ( y  =  A  ->  (
x  .o  y )  =  ( x  .o  A ) )
32mpteq2dv 4186 . . . . 5  |-  ( y  =  A  ->  (
x  e.  _V  |->  ( x  .o  y ) )  =  ( x  e.  _V  |->  ( x  .o  A ) ) )
4 rdgeq1 6508 . . . . 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 5607 . . 3  |-  ( y  =  A  ->  ( rec ( ( x  e. 
_V  |->  ( x  .o  y ) ) ,  1o ) `  z
)  =  ( rec ( ( x  e. 
_V  |->  ( x  .o  A ) ) ,  1o ) `  z
) )
71, 6ifbieq2d 3661 . 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 3364 . . 3  |-  ( z  =  B  ->  ( 1o  \  z )  =  ( 1o  \  B
) )
9 fveq2 5605 . . 3  |-  ( z  =  B  ->  ( rec ( ( x  e. 
_V  |->  ( x  .o  A ) ) ,  1o ) `  z
)  =  ( rec ( ( x  e. 
_V  |->  ( x  .o  A ) ) ,  1o ) `  B
) )
108, 9ifeq12d 3657 . 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 6569 . 2  |-  ^o  =  ( y  e.  On ,  z  e.  On  |->  if ( y  =  (/) ,  ( 1o  \  z
) ,  ( rec ( ( x  e. 
_V  |->  ( x  .o  y ) ) ,  1o ) `  z
) ) )
12 1on 6570 . . . . 5  |-  1o  e.  On
1312elexi 2873 . . . 4  |-  1o  e.  _V
14 difss 3379 . . . 4  |-  ( 1o 
\  B )  C_  1o
1513, 14ssexi 4238 . . 3  |-  ( 1o 
\  B )  e. 
_V
16 fvex 5619 . . 3  |-  ( rec ( ( x  e. 
_V  |->  ( x  .o  A ) ) ,  1o ) `  B
)  e.  _V
1715, 16ifex 3699 . 2  |-  if ( A  =  (/) ,  ( 1o  \  B ) ,  ( rec (
( x  e.  _V  |->  ( x  .o  A
) ) ,  1o ) `  B )
)  e.  _V
187, 10, 11, 17ovmpt2 6067 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 1642    e. wcel 1710   _Vcvv 2864    \ cdif 3225   (/)c0 3531   ifcif 3641    e. cmpt 4156   Oncon0 4471   ` cfv 5334  (class class class)co 5942   reccrdg 6506   1oc1o 6556    .o comu 6561    ^o coe 6562
This theorem is referenced by:  oevn0  6598  oe0m  6601
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-sep 4220  ax-nul 4228  ax-pr 4293  ax-un 4591
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-ral 2624  df-rex 2625  df-rab 2628  df-v 2866  df-sbc 3068  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-pss 3244  df-nul 3532  df-if 3642  df-pw 3703  df-sn 3722  df-pr 3723  df-tp 3724  df-op 3725  df-uni 3907  df-br 4103  df-opab 4157  df-mpt 4158  df-tr 4193  df-eprel 4384  df-id 4388  df-po 4393  df-so 4394  df-fr 4431  df-we 4433  df-ord 4474  df-on 4475  df-suc 4477  df-xp 4774  df-rel 4775  df-cnv 4776  df-co 4777  df-dm 4778  df-iota 5298  df-fun 5336  df-fv 5342  df-ov 5945  df-oprab 5946  df-mpt2 5947  df-recs 6472  df-rdg 6507  df-1o 6563  df-oexp 6569
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