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Theorem fparlem2 6672
Description: Lemma for fpar 6675. (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 5701 . . . . . 6  |-  ( x  e.  ( _V  X.  _V )  ->  ( ( 2nd  |`  ( _V  X.  _V ) ) `  x )  =  ( 2nd `  x ) )
21eqeq1d 2449 . . . . 5  |-  ( x  e.  ( _V  X.  _V )  ->  ( ( ( 2nd  |`  ( _V  X.  _V ) ) `
 x )  =  y  <->  ( 2nd `  x
)  =  y ) )
3 vex 2973 . . . . . . 7  |-  y  e. 
_V
43elsnc2 3905 . . . . . 6  |-  ( ( 2nd `  x )  e.  { y }  <-> 
( 2nd `  x
)  =  y )
5 fvex 5698 . . . . . . 7  |-  ( 1st `  x )  e.  _V
65biantrur 503 . . . . . 6  |-  ( ( 2nd `  x )  e.  { y }  <-> 
( ( 1st `  x
)  e.  _V  /\  ( 2nd `  x )  e.  { y } ) )
74, 6bitr3i 251 . . . . 5  |-  ( ( 2nd `  x )  =  y  <->  ( ( 1st `  x )  e. 
_V  /\  ( 2nd `  x )  e.  {
y } ) )
82, 7syl6bb 261 . . . 4  |-  ( x  e.  ( _V  X.  _V )  ->  ( ( ( 2nd  |`  ( _V  X.  _V ) ) `
 x )  =  y  <->  ( ( 1st `  x )  e.  _V  /\  ( 2nd `  x
)  e.  { y } ) ) )
98pm5.32i 632 . . 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 6598 . . . 4  |-  ( 2nd  |`  ( _V  X.  _V ) ) : ( _V  X.  _V ) --> _V
11 ffn 5556 . . . 4  |-  ( ( 2nd  |`  ( _V  X.  _V ) ) : ( _V  X.  _V )
--> _V  ->  ( 2nd  |`  ( _V  X.  _V ) )  Fn  ( _V  X.  _V ) )
12 fniniseg 5821 . . . 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 6608 . . 3  |-  ( x  e.  ( _V  X.  { y } )  <-> 
( x  e.  ( _V  X.  _V )  /\  ( ( 1st `  x
)  e.  _V  /\  ( 2nd `  x )  e.  { y } ) ) )
159, 13, 143bitr4i 277 . 2  |-  ( x  e.  ( `' ( 2nd  |`  ( _V  X.  _V ) ) " { y } )  <-> 
x  e.  ( _V 
X.  { y } ) )
1615eqriv 2438 1  |-  ( `' ( 2nd  |`  ( _V  X.  _V ) )
" { y } )  =  ( _V 
X.  { y } )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    /\ wa 369    = wceq 1364    e. wcel 1761   _Vcvv 2970   {csn 3874    X. cxp 4834   `'ccnv 4835    |` cres 4838   "cima 4839    Fn wfn 5410   -->wf 5411   ` cfv 5415   1stc1st 6574   2ndc2nd 6575
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2261  df-mo 2262  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-ral 2718  df-rex 2719  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-nul 3635  df-if 3789  df-sn 3875  df-pr 3877  df-op 3881  df-uni 4089  df-iun 4170  df-br 4290  df-opab 4348  df-mpt 4349  df-id 4632  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-fv 5423  df-1st 6576  df-2nd 6577
This theorem is referenced by:  fparlem4  6674
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