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Theorem pw2f1o2val 27110
Description: Function value of the pw2f1o2 27109 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 5405 . . 3  |-  ( X  e.  ( 2o  ^m  A )  ->  `' X  e.  _V )
2 imaexg 5217 . . 3  |-  ( `' X  e.  _V  ->  ( `' X " { 1o } )  e.  _V )
31, 2syl 16 . 2  |-  ( X  e.  ( 2o  ^m  A )  ->  ( `' X " { 1o } )  e.  _V )
4 cnveq 5046 . . . 4  |-  ( x  =  X  ->  `' x  =  `' X
)
54imaeq1d 5202 . . 3  |-  ( x  =  X  ->  ( `' x " { 1o } )  =  ( `' X " { 1o } ) )
6 pw2f1o2.f . . 3  |-  F  =  ( x  e.  ( 2o  ^m  A ) 
|->  ( `' x " { 1o } ) )
75, 6fvmptg 5804 . 2  |-  ( ( X  e.  ( 2o 
^m  A )  /\  ( `' X " { 1o } )  e.  _V )  ->  ( F `  X )  =  ( `' X " { 1o } ) )
83, 7mpdan 650 1  |-  ( X  e.  ( 2o  ^m  A )  ->  ( F `  X )  =  ( `' X " { 1o } ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1652    e. wcel 1725   _Vcvv 2956   {csn 3814    e. cmpt 4266   `'ccnv 4877   "cima 4881   ` cfv 5454  (class class class)co 6081   1oc1o 6717   2oc2o 6718    ^m cmap 7018
This theorem is referenced by:  pw2f1o2val2  27111
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-sbc 3162  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fv 5462
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