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Theorem oev 7067
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 2458 . . 3  |-  ( y  =  A  ->  (
y  =  (/)  <->  A  =  (/) ) )
2 oveq2 6211 . . . . . 6  |-  ( y  =  A  ->  (
x  .o  y )  =  ( x  .o  A ) )
32mpteq2dv 4490 . . . . 5  |-  ( y  =  A  ->  (
x  e.  _V  |->  ( x  .o  y ) )  =  ( x  e.  _V  |->  ( x  .o  A ) ) )
4 rdgeq1 6980 . . . . 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 5804 . . 3  |-  ( y  =  A  ->  ( rec ( ( x  e. 
_V  |->  ( x  .o  y ) ) ,  1o ) `  z
)  =  ( rec ( ( x  e. 
_V  |->  ( x  .o  A ) ) ,  1o ) `  z
) )
71, 6ifbieq2d 3925 . 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 3579 . . 3  |-  ( z  =  B  ->  ( 1o  \  z )  =  ( 1o  \  B
) )
9 fveq2 5802 . . 3  |-  ( z  =  B  ->  ( rec ( ( x  e. 
_V  |->  ( x  .o  A ) ) ,  1o ) `  z
)  =  ( rec ( ( x  e. 
_V  |->  ( x  .o  A ) ) ,  1o ) `  B
) )
108, 9ifeq12d 3920 . 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 7039 . 2  |-  ^o  =  ( y  e.  On ,  z  e.  On  |->  if ( y  =  (/) ,  ( 1o  \  z
) ,  ( rec ( ( x  e. 
_V  |->  ( x  .o  y ) ) ,  1o ) `  z
) ) )
12 1on 7040 . . . . 5  |-  1o  e.  On
1312elexi 3088 . . . 4  |-  1o  e.  _V
14 difss 3594 . . . 4  |-  ( 1o 
\  B )  C_  1o
1513, 14ssexi 4548 . . 3  |-  ( 1o 
\  B )  e. 
_V
16 fvex 5812 . . 3  |-  ( rec ( ( x  e. 
_V  |->  ( x  .o  A ) ) ,  1o ) `  B
)  e.  _V
1715, 16ifex 3969 . 2  |-  if ( A  =  (/) ,  ( 1o  \  B ) ,  ( rec (
( x  e.  _V  |->  ( x  .o  A
) ) ,  1o ) `  B )
)  e.  _V
187, 10, 11, 17ovmpt2 6339 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 1370    e. wcel 1758   _Vcvv 3078    \ cdif 3436   (/)c0 3748   ifcif 3902    |-> cmpt 4461   Oncon0 4830   ` cfv 5529  (class class class)co 6203   reccrdg 6978   1oc1o 7026    .o comu 7031    ^o coe 7032
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4524  ax-nul 4532  ax-pr 4642  ax-un 6485
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-rab 2808  df-v 3080  df-sbc 3295  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-pss 3455  df-nul 3749  df-if 3903  df-pw 3973  df-sn 3989  df-pr 3991  df-tp 3993  df-op 3995  df-uni 4203  df-br 4404  df-opab 4462  df-mpt 4463  df-tr 4497  df-eprel 4743  df-id 4747  df-po 4752  df-so 4753  df-fr 4790  df-we 4792  df-ord 4833  df-on 4834  df-suc 4836  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-iota 5492  df-fun 5531  df-fv 5537  df-ov 6206  df-oprab 6207  df-mpt2 6208  df-recs 6945  df-rdg 6979  df-1o 7033  df-oexp 7039
This theorem is referenced by:  oevn0  7068  oe0m  7071
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