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Theorem fparlem1 6873
Description: Lemma for fpar 6877. (Contributed by NM, 22-Dec-2008.) (Revised by Mario Carneiro, 28-Apr-2015.)
Assertion
Ref Expression
fparlem1  |-  ( `' ( 1st  |`  ( _V  X.  _V ) )
" { x }
)  =  ( { x }  X.  _V )

Proof of Theorem fparlem1
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 fvres 5862 . . . . . 6  |-  ( y  e.  ( _V  X.  _V )  ->  ( ( 1st  |`  ( _V  X.  _V ) ) `  y )  =  ( 1st `  y ) )
21eqeq1d 2456 . . . . 5  |-  ( y  e.  ( _V  X.  _V )  ->  ( ( ( 1st  |`  ( _V  X.  _V ) ) `
 y )  =  x  <->  ( 1st `  y
)  =  x ) )
3 vex 3109 . . . . . . 7  |-  x  e. 
_V
43elsnc2 4047 . . . . . 6  |-  ( ( 1st `  y )  e.  { x }  <->  ( 1st `  y )  =  x )
5 fvex 5858 . . . . . . 7  |-  ( 2nd `  y )  e.  _V
65biantru 503 . . . . . 6  |-  ( ( 1st `  y )  e.  { x }  <->  ( ( 1st `  y
)  e.  { x }  /\  ( 2nd `  y
)  e.  _V )
)
74, 6bitr3i 251 . . . . 5  |-  ( ( 1st `  y )  =  x  <->  ( ( 1st `  y )  e. 
{ x }  /\  ( 2nd `  y )  e.  _V ) )
82, 7syl6bb 261 . . . 4  |-  ( y  e.  ( _V  X.  _V )  ->  ( ( ( 1st  |`  ( _V  X.  _V ) ) `
 y )  =  x  <->  ( ( 1st `  y )  e.  {
x }  /\  ( 2nd `  y )  e. 
_V ) ) )
98pm5.32i 635 . . 3  |-  ( ( y  e.  ( _V 
X.  _V )  /\  (
( 1st  |`  ( _V 
X.  _V ) ) `  y )  =  x )  <->  ( y  e.  ( _V  X.  _V )  /\  ( ( 1st `  y )  e.  {
x }  /\  ( 2nd `  y )  e. 
_V ) ) )
10 f1stres 6795 . . . 4  |-  ( 1st  |`  ( _V  X.  _V ) ) : ( _V  X.  _V ) --> _V
11 ffn 5713 . . . 4  |-  ( ( 1st  |`  ( _V  X.  _V ) ) : ( _V  X.  _V )
--> _V  ->  ( 1st  |`  ( _V  X.  _V ) )  Fn  ( _V  X.  _V ) )
12 fniniseg 5984 . . . 4  |-  ( ( 1st  |`  ( _V  X.  _V ) )  Fn  ( _V  X.  _V )  ->  ( y  e.  ( `' ( 1st  |`  ( _V  X.  _V ) ) " {
x } )  <->  ( y  e.  ( _V  X.  _V )  /\  ( ( 1st  |`  ( _V  X.  _V ) ) `  y
)  =  x ) ) )
1310, 11, 12mp2b 10 . . 3  |-  ( y  e.  ( `' ( 1st  |`  ( _V  X.  _V ) ) " { x } )  <-> 
( y  e.  ( _V  X.  _V )  /\  ( ( 1st  |`  ( _V  X.  _V ) ) `
 y )  =  x ) )
14 elxp7 6806 . . 3  |-  ( y  e.  ( { x }  X.  _V )  <->  ( y  e.  ( _V  X.  _V )  /\  ( ( 1st `  y )  e.  {
x }  /\  ( 2nd `  y )  e. 
_V ) ) )
159, 13, 143bitr4i 277 . 2  |-  ( y  e.  ( `' ( 1st  |`  ( _V  X.  _V ) ) " { x } )  <-> 
y  e.  ( { x }  X.  _V ) )
1615eqriv 2450 1  |-  ( `' ( 1st  |`  ( _V  X.  _V ) )
" { x }
)  =  ( { x }  X.  _V )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    /\ wa 367    = wceq 1398    e. wcel 1823   _Vcvv 3106   {csn 4016    X. cxp 4986   `'ccnv 4987    |` cres 4990   "cima 4991    Fn wfn 5565   -->wf 5566   ` cfv 5570   1stc1st 6771   2ndc2nd 6772
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-id 4784  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-fv 5578  df-1st 6773  df-2nd 6774
This theorem is referenced by:  fparlem3  6875
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