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Theorem pw2f1o2val 29531
Description: Function value of the pw2f1o2 29530 bijection. (Contributed by Stefan O'Rear, 18-Jan-2015.) (Revised by Stefan O'Rear, 6-May-2015.)
Hypothesis
Ref Expression
pw2f1o2.f  |-  F  =  ( x  e.  ( 2o  ^m  A ) 
|->  ( `' x " { 1o } ) )
Assertion
Ref Expression
pw2f1o2val  |-  ( X  e.  ( 2o  ^m  A )  ->  ( F `  X )  =  ( `' X " { 1o } ) )
Distinct variable groups:    x, A    x, X
Allowed substitution hint:    F( x)

Proof of Theorem pw2f1o2val
StepHypRef Expression
1 cnvexg 6629 . . 3  |-  ( X  e.  ( 2o  ^m  A )  ->  `' X  e.  _V )
2 imaexg 6620 . . 3  |-  ( `' X  e.  _V  ->  ( `' X " { 1o } )  e.  _V )
31, 2syl 16 . 2  |-  ( X  e.  ( 2o  ^m  A )  ->  ( `' X " { 1o } )  e.  _V )
4 cnveq 5116 . . . 4  |-  ( x  =  X  ->  `' x  =  `' X
)
54imaeq1d 5271 . . 3  |-  ( x  =  X  ->  ( `' x " { 1o } )  =  ( `' X " { 1o } ) )
6 pw2f1o2.f . . 3  |-  F  =  ( x  e.  ( 2o  ^m  A ) 
|->  ( `' x " { 1o } ) )
75, 6fvmptg 5876 . 2  |-  ( ( X  e.  ( 2o 
^m  A )  /\  ( `' X " { 1o } )  e.  _V )  ->  ( F `  X )  =  ( `' X " { 1o } ) )
83, 7mpdan 668 1  |-  ( X  e.  ( 2o  ^m  A )  ->  ( F `  X )  =  ( `' X " { 1o } ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1370    e. wcel 1758   _Vcvv 3072   {csn 3980    |-> cmpt 4453   `'ccnv 4942   "cima 4946   ` cfv 5521  (class class class)co 6195   1oc1o 7018   2oc2o 7019    ^m cmap 7319
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 1954  ax-ext 2431  ax-sep 4516  ax-nul 4524  ax-pow 4573  ax-pr 4634  ax-un 6477
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2265  df-mo 2266  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2602  df-ne 2647  df-ral 2801  df-rex 2802  df-rab 2805  df-v 3074  df-sbc 3289  df-dif 3434  df-un 3436  df-in 3438  df-ss 3445  df-nul 3741  df-if 3895  df-pw 3965  df-sn 3981  df-pr 3983  df-op 3987  df-uni 4195  df-br 4396  df-opab 4454  df-mpt 4455  df-id 4739  df-xp 4949  df-rel 4950  df-cnv 4951  df-co 4952  df-dm 4953  df-rn 4954  df-res 4955  df-ima 4956  df-iota 5484  df-fun 5523  df-fv 5529
This theorem is referenced by:  pw2f1o2val2  29532
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