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Theorem fparlem2 6449
Description: Lemma for fpar 6452. (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 5747 . . . . . 6  |-  ( x  e.  ( _V  X.  _V )  ->  ( ( 2nd  |`  ( _V  X.  _V ) ) `  x )  =  ( 2nd `  x ) )
21eqeq1d 2446 . . . . 5  |-  ( x  e.  ( _V  X.  _V )  ->  ( ( ( 2nd  |`  ( _V  X.  _V ) ) `
 x )  =  y  <->  ( 2nd `  x
)  =  y ) )
3 vex 2961 . . . . . . 7  |-  y  e. 
_V
43elsnc2 3845 . . . . . 6  |-  ( ( 2nd `  x )  e.  { y }  <-> 
( 2nd `  x
)  =  y )
5 fvex 5744 . . . . . . 7  |-  ( 1st `  x )  e.  _V
65biantrur 494 . . . . . 6  |-  ( ( 2nd `  x )  e.  { y }  <-> 
( ( 1st `  x
)  e.  _V  /\  ( 2nd `  x )  e.  { y } ) )
74, 6bitr3i 244 . . . . 5  |-  ( ( 2nd `  x )  =  y  <->  ( ( 1st `  x )  e. 
_V  /\  ( 2nd `  x )  e.  {
y } ) )
82, 7syl6bb 254 . . . 4  |-  ( x  e.  ( _V  X.  _V )  ->  ( ( ( 2nd  |`  ( _V  X.  _V ) ) `
 x )  =  y  <->  ( ( 1st `  x )  e.  _V  /\  ( 2nd `  x
)  e.  { y } ) ) )
98pm5.32i 620 . . 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 6371 . . . 4  |-  ( 2nd  |`  ( _V  X.  _V ) ) : ( _V  X.  _V ) --> _V
11 ffn 5593 . . . 4  |-  ( ( 2nd  |`  ( _V  X.  _V ) ) : ( _V  X.  _V )
--> _V  ->  ( 2nd  |`  ( _V  X.  _V ) )  Fn  ( _V  X.  _V ) )
12 fniniseg 5853 . . . 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 10 . . 3  |-  ( x  e.  ( `' ( 2nd  |`  ( _V  X.  _V ) ) " { y } )  <-> 
( x  e.  ( _V  X.  _V )  /\  ( ( 2nd  |`  ( _V  X.  _V ) ) `
 x )  =  y ) )
14 elxp7 6381 . . 3  |-  ( x  e.  ( _V  X.  { y } )  <-> 
( x  e.  ( _V  X.  _V )  /\  ( ( 1st `  x
)  e.  _V  /\  ( 2nd `  x )  e.  { y } ) ) )
159, 13, 143bitr4i 270 . 2  |-  ( x  e.  ( `' ( 2nd  |`  ( _V  X.  _V ) ) " { y } )  <-> 
x  e.  ( _V 
X.  { y } ) )
1615eqriv 2435 1  |-  ( `' ( 2nd  |`  ( _V  X.  _V ) )
" { y } )  =  ( _V 
X.  { y } )
Colors of variables: wff set class
Syntax hints:    <-> wb 178    /\ wa 360    = wceq 1653    e. wcel 1726   _Vcvv 2958   {csn 3816    X. cxp 4878   `'ccnv 4879    |` cres 4882   "cima 4883    Fn wfn 5451   -->wf 5452   ` cfv 5456   1stc1st 6349   2ndc2nd 6350
This theorem is referenced by:  fparlem4  6451
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-iun 4097  df-br 4215  df-opab 4269  df-mpt 4270  df-id 4500  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-fv 5464  df-1st 6351  df-2nd 6352
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