Users' Mathboxes Mathbox for Stefan O'Rear < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  pw2f1o2val Unicode version

Theorem pw2f1o2val 27132
Description: Function value of the pw2f1o2 27131 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 5208 . . 3  |-  ( X  e.  ( 2o  ^m  A )  ->  `' X  e.  _V )
2 imaexg 5026 . . 3  |-  ( `' X  e.  _V  ->  ( `' X " { 1o } )  e.  _V )
31, 2syl 15 . 2  |-  ( X  e.  ( 2o  ^m  A )  ->  ( `' X " { 1o } )  e.  _V )
4 cnveq 4855 . . . 4  |-  ( x  =  X  ->  `' x  =  `' X
)
54imaeq1d 5011 . . 3  |-  ( x  =  X  ->  ( `' x " { 1o } )  =  ( `' X " { 1o } ) )
6 pw2f1o2.f . . 3  |-  F  =  ( x  e.  ( 2o  ^m  A ) 
|->  ( `' x " { 1o } ) )
75, 6fvmptg 5600 . 2  |-  ( ( X  e.  ( 2o 
^m  A )  /\  ( `' X " { 1o } )  e.  _V )  ->  ( F `  X )  =  ( `' X " { 1o } ) )
83, 7mpdan 649 1  |-  ( X  e.  ( 2o  ^m  A )  ->  ( F `  X )  =  ( `' X " { 1o } ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1623    e. wcel 1684   _Vcvv 2788   {csn 3640    e. cmpt 4077   `'ccnv 4688   "cima 4692   ` cfv 5255  (class class class)co 5858   1oc1o 6472   2oc2o 6473    ^m cmap 6772
This theorem is referenced by:  pw2f1o2val2  27133
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fv 5263
  Copyright terms: Public domain W3C validator