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Theorem fparlem1 6413
Description: Lemma for fpar 6417. (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 5712 . . . . . 6  |-  ( y  e.  ( _V  X.  _V )  ->  ( ( 1st  |`  ( _V  X.  _V ) ) `  y )  =  ( 1st `  y ) )
21eqeq1d 2420 . . . . 5  |-  ( y  e.  ( _V  X.  _V )  ->  ( ( ( 1st  |`  ( _V  X.  _V ) ) `
 y )  =  x  <->  ( 1st `  y
)  =  x ) )
3 vex 2927 . . . . . . 7  |-  x  e. 
_V
43elsnc2 3811 . . . . . 6  |-  ( ( 1st `  y )  e.  { x }  <->  ( 1st `  y )  =  x )
5 fvex 5709 . . . . . . 7  |-  ( 2nd `  y )  e.  _V
65biantru 492 . . . . . 6  |-  ( ( 1st `  y )  e.  { x }  <->  ( ( 1st `  y
)  e.  { x }  /\  ( 2nd `  y
)  e.  _V )
)
74, 6bitr3i 243 . . . . 5  |-  ( ( 1st `  y )  =  x  <->  ( ( 1st `  y )  e. 
{ x }  /\  ( 2nd `  y )  e.  _V ) )
82, 7syl6bb 253 . . . 4  |-  ( y  e.  ( _V  X.  _V )  ->  ( ( ( 1st  |`  ( _V  X.  _V ) ) `
 y )  =  x  <->  ( ( 1st `  y )  e.  {
x }  /\  ( 2nd `  y )  e. 
_V ) ) )
98pm5.32i 619 . . 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 6335 . . . 4  |-  ( 1st  |`  ( _V  X.  _V ) ) : ( _V  X.  _V ) --> _V
11 ffn 5558 . . . 4  |-  ( ( 1st  |`  ( _V  X.  _V ) ) : ( _V  X.  _V )
--> _V  ->  ( 1st  |`  ( _V  X.  _V ) )  Fn  ( _V  X.  _V ) )
12 fniniseg 5818 . . . 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 6346 . . 3  |-  ( y  e.  ( { x }  X.  _V )  <->  ( y  e.  ( _V  X.  _V )  /\  ( ( 1st `  y )  e.  {
x }  /\  ( 2nd `  y )  e. 
_V ) ) )
159, 13, 143bitr4i 269 . 2  |-  ( y  e.  ( `' ( 1st  |`  ( _V  X.  _V ) ) " { x } )  <-> 
y  e.  ( { x }  X.  _V ) )
1615eqriv 2409 1  |-  ( `' ( 1st  |`  ( _V  X.  _V ) )
" { x }
)  =  ( { x }  X.  _V )
Colors of variables: wff set class
Syntax hints:    <-> wb 177    /\ wa 359    = wceq 1649    e. wcel 1721   _Vcvv 2924   {csn 3782    X. cxp 4843   `'ccnv 4844    |` cres 4847   "cima 4848    Fn wfn 5416   -->wf 5417   ` cfv 5421   1stc1st 6314   2ndc2nd 6315
This theorem is referenced by:  fparlem3  6415
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393  ax-sep 4298  ax-nul 4306  ax-pow 4345  ax-pr 4371  ax-un 4668
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-ral 2679  df-rex 2680  df-rab 2683  df-v 2926  df-sbc 3130  df-csb 3220  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-nul 3597  df-if 3708  df-sn 3788  df-pr 3789  df-op 3791  df-uni 3984  df-iun 4063  df-br 4181  df-opab 4235  df-mpt 4236  df-id 4466  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5385  df-fun 5423  df-fn 5424  df-f 5425  df-fv 5429  df-1st 6316  df-2nd 6317
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