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Theorem fparlem2 6876
Description: Lemma for fpar 6879. (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 5873 . . . . . 6  |-  ( x  e.  ( _V  X.  _V )  ->  ( ( 2nd  |`  ( _V  X.  _V ) ) `  x )  =  ( 2nd `  x ) )
21eqeq1d 2464 . . . . 5  |-  ( x  e.  ( _V  X.  _V )  ->  ( ( ( 2nd  |`  ( _V  X.  _V ) ) `
 x )  =  y  <->  ( 2nd `  x
)  =  y ) )
3 vex 3111 . . . . . . 7  |-  y  e. 
_V
43elsnc2 4053 . . . . . 6  |-  ( ( 2nd `  x )  e.  { y }  <-> 
( 2nd `  x
)  =  y )
5 fvex 5869 . . . . . . 7  |-  ( 1st `  x )  e.  _V
65biantrur 506 . . . . . 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 637 . . 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 6799 . . . 4  |-  ( 2nd  |`  ( _V  X.  _V ) ) : ( _V  X.  _V ) --> _V
11 ffn 5724 . . . 4  |-  ( ( 2nd  |`  ( _V  X.  _V ) ) : ( _V  X.  _V )
--> _V  ->  ( 2nd  |`  ( _V  X.  _V ) )  Fn  ( _V  X.  _V ) )
12 fniniseg 5995 . . . 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 6809 . . 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 2458 1  |-  ( `' ( 2nd  |`  ( _V  X.  _V ) )
" { y } )  =  ( _V 
X.  { y } )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    /\ wa 369    = wceq 1374    e. wcel 1762   _Vcvv 3108   {csn 4022    X. cxp 4992   `'ccnv 4993    |` cres 4996   "cima 4997    Fn wfn 5576   -->wf 5577   ` cfv 5581   1stc1st 6774   2ndc2nd 6775
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 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1963  ax-ext 2440  ax-sep 4563  ax-nul 4571  ax-pow 4620  ax-pr 4681  ax-un 6569
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2274  df-mo 2275  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2612  df-ne 2659  df-ral 2814  df-rex 2815  df-rab 2818  df-v 3110  df-sbc 3327  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3781  df-if 3935  df-sn 4023  df-pr 4025  df-op 4029  df-uni 4241  df-iun 4322  df-br 4443  df-opab 4501  df-mpt 4502  df-id 4790  df-xp 5000  df-rel 5001  df-cnv 5002  df-co 5003  df-dm 5004  df-rn 5005  df-res 5006  df-ima 5007  df-iota 5544  df-fun 5583  df-fn 5584  df-f 5585  df-fv 5589  df-1st 6776  df-2nd 6777
This theorem is referenced by:  fparlem4  6878
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