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Theorem pw2f1o2val 31223
Description: Function value of the pw2f1o2 31222 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 6719 . . 3  |-  ( X  e.  ( 2o  ^m  A )  ->  `' X  e.  _V )
2 imaexg 6710 . . 3  |-  ( `' X  e.  _V  ->  ( `' X " { 1o } )  e.  _V )
31, 2syl 16 . 2  |-  ( X  e.  ( 2o  ^m  A )  ->  ( `' X " { 1o } )  e.  _V )
4 cnveq 5165 . . . 4  |-  ( x  =  X  ->  `' x  =  `' X
)
54imaeq1d 5324 . . 3  |-  ( x  =  X  ->  ( `' x " { 1o } )  =  ( `' X " { 1o } ) )
6 pw2f1o2.f . . 3  |-  F  =  ( x  e.  ( 2o  ^m  A ) 
|->  ( `' x " { 1o } ) )
75, 6fvmptg 5929 . 2  |-  ( ( X  e.  ( 2o 
^m  A )  /\  ( `' X " { 1o } )  e.  _V )  ->  ( F `  X )  =  ( `' X " { 1o } ) )
83, 7mpdan 666 1  |-  ( X  e.  ( 2o  ^m  A )  ->  ( F `  X )  =  ( `' X " { 1o } ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1398    e. wcel 1823   _Vcvv 3106   {csn 4016    |-> cmpt 4497   `'ccnv 4987   "cima 4991   ` cfv 5570  (class class class)co 6270   1oc1o 7115   2oc2o 7116    ^m cmap 7412
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-sbc 3325  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-br 4440  df-opab 4498  df-mpt 4499  df-id 4784  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fv 5578
This theorem is referenced by:  pw2f1o2val2  31224
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