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Theorem fparlem2 6235
Description: Lemma for fpar 6238. (Contributed by NM, 22-Dec-2008.) (Revised by Mario Carneiro, 28-Apr-2015.)
Assertion
Ref Expression
fparlem2  |-  ( `' ( 2nd  |`  ( _V  X.  _V ) )
" { y } )  =  ( _V 
X.  { y } )

Proof of Theorem fparlem2
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 fvres 5558 . . . . . 6  |-  ( x  e.  ( _V  X.  _V )  ->  ( ( 2nd  |`  ( _V  X.  _V ) ) `  x )  =  ( 2nd `  x ) )
21eqeq1d 2304 . . . . 5  |-  ( x  e.  ( _V  X.  _V )  ->  ( ( ( 2nd  |`  ( _V  X.  _V ) ) `
 x )  =  y  <->  ( 2nd `  x
)  =  y ) )
3 vex 2804 . . . . . . 7  |-  y  e. 
_V
43elsnc2 3682 . . . . . 6  |-  ( ( 2nd `  x )  e.  { y }  <-> 
( 2nd `  x
)  =  y )
5 fvex 5555 . . . . . . 7  |-  ( 1st `  x )  e.  _V
65biantrur 492 . . . . . 6  |-  ( ( 2nd `  x )  e.  { y }  <-> 
( ( 1st `  x
)  e.  _V  /\  ( 2nd `  x )  e.  { y } ) )
74, 6bitr3i 242 . . . . 5  |-  ( ( 2nd `  x )  =  y  <->  ( ( 1st `  x )  e. 
_V  /\  ( 2nd `  x )  e.  {
y } ) )
82, 7syl6bb 252 . . . 4  |-  ( x  e.  ( _V  X.  _V )  ->  ( ( ( 2nd  |`  ( _V  X.  _V ) ) `
 x )  =  y  <->  ( ( 1st `  x )  e.  _V  /\  ( 2nd `  x
)  e.  { y } ) ) )
98pm5.32i 618 . . 3  |-  ( ( x  e.  ( _V 
X.  _V )  /\  (
( 2nd  |`  ( _V 
X.  _V ) ) `  x )  =  y )  <->  ( x  e.  ( _V  X.  _V )  /\  ( ( 1st `  x )  e.  _V  /\  ( 2nd `  x
)  e.  { y } ) ) )
10 f2ndres 6158 . . . 4  |-  ( 2nd  |`  ( _V  X.  _V ) ) : ( _V  X.  _V ) --> _V
11 ffn 5405 . . . 4  |-  ( ( 2nd  |`  ( _V  X.  _V ) ) : ( _V  X.  _V )
--> _V  ->  ( 2nd  |`  ( _V  X.  _V ) )  Fn  ( _V  X.  _V ) )
12 fniniseg 5662 . . . 4  |-  ( ( 2nd  |`  ( _V  X.  _V ) )  Fn  ( _V  X.  _V )  ->  ( x  e.  ( `' ( 2nd  |`  ( _V  X.  _V ) ) " {
y } )  <->  ( x  e.  ( _V  X.  _V )  /\  ( ( 2nd  |`  ( _V  X.  _V ) ) `  x
)  =  y ) ) )
1310, 11, 12mp2b 9 . . 3  |-  ( x  e.  ( `' ( 2nd  |`  ( _V  X.  _V ) ) " { y } )  <-> 
( x  e.  ( _V  X.  _V )  /\  ( ( 2nd  |`  ( _V  X.  _V ) ) `
 x )  =  y ) )
14 elxp7 6168 . . 3  |-  ( x  e.  ( _V  X.  { y } )  <-> 
( x  e.  ( _V  X.  _V )  /\  ( ( 1st `  x
)  e.  _V  /\  ( 2nd `  x )  e.  { y } ) ) )
159, 13, 143bitr4i 268 . 2  |-  ( x  e.  ( `' ( 2nd  |`  ( _V  X.  _V ) ) " { y } )  <-> 
x  e.  ( _V 
X.  { y } ) )
1615eqriv 2293 1  |-  ( `' ( 2nd  |`  ( _V  X.  _V ) )
" { y } )  =  ( _V 
X.  { y } )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    /\ wa 358    = wceq 1632    e. wcel 1696   _Vcvv 2801   {csn 3653    X. cxp 4703   `'ccnv 4704    |` cres 4707   "cima 4708    Fn wfn 5266   -->wf 5267   ` cfv 5271   1stc1st 6136   2ndc2nd 6137
This theorem is referenced by:  fparlem4  6237
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-fv 5279  df-1st 6138  df-2nd 6139
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