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Theorem oev 7171
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 2432 . . 3  |-  ( y  =  A  ->  (
y  =  (/)  <->  A  =  (/) ) )
2 oveq2 6257 . . . . . 6  |-  ( y  =  A  ->  (
x  .o  y )  =  ( x  .o  A ) )
32mpteq2dv 4454 . . . . 5  |-  ( y  =  A  ->  (
x  e.  _V  |->  ( x  .o  y ) )  =  ( x  e.  _V  |->  ( x  .o  A ) ) )
4 rdgeq1 7084 . . . . 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 17 . . . 4  |-  ( y  =  A  ->  rec ( ( x  e. 
_V  |->  ( x  .o  y ) ) ,  1o )  =  rec ( ( x  e. 
_V  |->  ( x  .o  A ) ) ,  1o ) )
65fveq1d 5827 . . 3  |-  ( y  =  A  ->  ( rec ( ( x  e. 
_V  |->  ( x  .o  y ) ) ,  1o ) `  z
)  =  ( rec ( ( x  e. 
_V  |->  ( x  .o  A ) ) ,  1o ) `  z
) )
71, 6ifbieq2d 3879 . 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 3520 . . 3  |-  ( z  =  B  ->  ( 1o  \  z )  =  ( 1o  \  B
) )
9 fveq2 5825 . . 3  |-  ( z  =  B  ->  ( rec ( ( x  e. 
_V  |->  ( x  .o  A ) ) ,  1o ) `  z
)  =  ( rec ( ( x  e. 
_V  |->  ( x  .o  A ) ) ,  1o ) `  B
) )
108, 9ifeq12d 3874 . 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 7143 . 2  |-  ^o  =  ( y  e.  On ,  z  e.  On  |->  if ( y  =  (/) ,  ( 1o  \  z
) ,  ( rec ( ( x  e. 
_V  |->  ( x  .o  y ) ) ,  1o ) `  z
) ) )
12 1on 7144 . . . . 5  |-  1o  e.  On
1312elexi 3032 . . . 4  |-  1o  e.  _V
14 difss 3535 . . . 4  |-  ( 1o 
\  B )  C_  1o
1513, 14ssexi 4512 . . 3  |-  ( 1o 
\  B )  e. 
_V
16 fvex 5835 . . 3  |-  ( rec ( ( x  e. 
_V  |->  ( x  .o  A ) ) ,  1o ) `  B
)  e.  _V
1715, 16ifex 3922 . 2  |-  if ( A  =  (/) ,  ( 1o  \  B ) ,  ( rec (
( x  e.  _V  |->  ( x  .o  A
) ) ,  1o ) `  B )
)  e.  _V
187, 10, 11, 17ovmpt2 6390 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 370    = wceq 1437    e. wcel 1872   _Vcvv 3022    \ cdif 3376   (/)c0 3704   ifcif 3854    |-> cmpt 4425   Oncon0 5385   ` cfv 5544  (class class class)co 6249   reccrdg 7082   1oc1o 7130    .o comu 7135    ^o coe 7136
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-8 1874  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2063  ax-ext 2408  ax-sep 4489  ax-nul 4498  ax-pr 4603  ax-un 6541
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2280  df-mo 2281  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2558  df-ne 2601  df-ral 2719  df-rex 2720  df-rab 2723  df-v 3024  df-sbc 3243  df-dif 3382  df-un 3384  df-in 3386  df-ss 3393  df-pss 3395  df-nul 3705  df-if 3855  df-pw 3926  df-sn 3942  df-pr 3944  df-tp 3946  df-op 3948  df-uni 4163  df-br 4367  df-opab 4426  df-mpt 4427  df-tr 4462  df-eprel 4707  df-id 4711  df-po 4717  df-so 4718  df-fr 4755  df-we 4757  df-xp 4802  df-rel 4803  df-cnv 4804  df-co 4805  df-dm 4806  df-rn 4807  df-res 4808  df-ima 4809  df-pred 5342  df-ord 5388  df-on 5389  df-suc 5391  df-iota 5508  df-fun 5546  df-fv 5552  df-ov 6252  df-oprab 6253  df-mpt2 6254  df-wrecs 6983  df-recs 7045  df-rdg 7083  df-1o 7137  df-oexp 7143
This theorem is referenced by:  oevn0  7172  oe0m  7175
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