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Theorem fparlem2 6839
Description: Lemma for fpar 6842. (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 5819 . . . . . 6  |-  ( x  e.  ( _V  X.  _V )  ->  ( ( 2nd  |`  ( _V  X.  _V ) ) `  x )  =  ( 2nd `  x ) )
21eqeq1d 2404 . . . . 5  |-  ( x  e.  ( _V  X.  _V )  ->  ( ( ( 2nd  |`  ( _V  X.  _V ) ) `
 x )  =  y  <->  ( 2nd `  x
)  =  y ) )
3 vex 3061 . . . . . . 7  |-  y  e. 
_V
43elsnc2 4002 . . . . . 6  |-  ( ( 2nd `  x )  e.  { y }  <-> 
( 2nd `  x
)  =  y )
5 fvex 5815 . . . . . . 7  |-  ( 1st `  x )  e.  _V
65biantrur 504 . . . . . 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 635 . . 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 6761 . . . 4  |-  ( 2nd  |`  ( _V  X.  _V ) ) : ( _V  X.  _V ) --> _V
11 ffn 5670 . . . 4  |-  ( ( 2nd  |`  ( _V  X.  _V ) ) : ( _V  X.  _V )
--> _V  ->  ( 2nd  |`  ( _V  X.  _V ) )  Fn  ( _V  X.  _V ) )
12 fniniseg 5942 . . . 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 6771 . . 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 2398 1  |-  ( `' ( 2nd  |`  ( _V  X.  _V ) )
" { y } )  =  ( _V 
X.  { y } )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    /\ wa 367    = wceq 1405    e. wcel 1842   _Vcvv 3058   {csn 3971    X. cxp 4940   `'ccnv 4941    |` cres 4944   "cima 4945    Fn wfn 5520   -->wf 5521   ` cfv 5525   1stc1st 6736   2ndc2nd 6737
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-sep 4516  ax-nul 4524  ax-pow 4571  ax-pr 4629  ax-un 6530
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2758  df-rex 2759  df-rab 2762  df-v 3060  df-sbc 3277  df-csb 3373  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-nul 3738  df-if 3885  df-sn 3972  df-pr 3974  df-op 3978  df-uni 4191  df-iun 4272  df-br 4395  df-opab 4453  df-mpt 4454  df-id 4737  df-xp 4948  df-rel 4949  df-cnv 4950  df-co 4951  df-dm 4952  df-rn 4953  df-res 4954  df-ima 4955  df-iota 5489  df-fun 5527  df-fn 5528  df-f 5529  df-fv 5533  df-1st 6738  df-2nd 6739
This theorem is referenced by:  fparlem4  6841
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