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Theorem fparlem1 6218
Description: Lemma for fpar 6222. (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 5542 . . . . . 6  |-  ( y  e.  ( _V  X.  _V )  ->  ( ( 1st  |`  ( _V  X.  _V ) ) `  y )  =  ( 1st `  y ) )
21eqeq1d 2291 . . . . 5  |-  ( y  e.  ( _V  X.  _V )  ->  ( ( ( 1st  |`  ( _V  X.  _V ) ) `
 y )  =  x  <->  ( 1st `  y
)  =  x ) )
3 vex 2791 . . . . . . 7  |-  x  e. 
_V
43elsnc2 3669 . . . . . 6  |-  ( ( 1st `  y )  e.  { x }  <->  ( 1st `  y )  =  x )
5 fvex 5539 . . . . . . 7  |-  ( 2nd `  y )  e.  _V
65biantru 491 . . . . . 6  |-  ( ( 1st `  y )  e.  { x }  <->  ( ( 1st `  y
)  e.  { x }  /\  ( 2nd `  y
)  e.  _V )
)
74, 6bitr3i 242 . . . . 5  |-  ( ( 1st `  y )  =  x  <->  ( ( 1st `  y )  e. 
{ x }  /\  ( 2nd `  y )  e.  _V ) )
82, 7syl6bb 252 . . . 4  |-  ( y  e.  ( _V  X.  _V )  ->  ( ( ( 1st  |`  ( _V  X.  _V ) ) `
 y )  =  x  <->  ( ( 1st `  y )  e.  {
x }  /\  ( 2nd `  y )  e. 
_V ) ) )
98pm5.32i 618 . . 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 6141 . . . 4  |-  ( 1st  |`  ( _V  X.  _V ) ) : ( _V  X.  _V ) --> _V
11 ffn 5389 . . . 4  |-  ( ( 1st  |`  ( _V  X.  _V ) ) : ( _V  X.  _V )
--> _V  ->  ( 1st  |`  ( _V  X.  _V ) )  Fn  ( _V  X.  _V ) )
12 fniniseg 5646 . . . 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 9 . . 3  |-  ( y  e.  ( `' ( 1st  |`  ( _V  X.  _V ) ) " { x } )  <-> 
( y  e.  ( _V  X.  _V )  /\  ( ( 1st  |`  ( _V  X.  _V ) ) `
 y )  =  x ) )
14 elxp7 6152 . . 3  |-  ( y  e.  ( { x }  X.  _V )  <->  ( y  e.  ( _V  X.  _V )  /\  ( ( 1st `  y )  e.  {
x }  /\  ( 2nd `  y )  e. 
_V ) ) )
159, 13, 143bitr4i 268 . 2  |-  ( y  e.  ( `' ( 1st  |`  ( _V  X.  _V ) ) " { x } )  <-> 
y  e.  ( { x }  X.  _V ) )
1615eqriv 2280 1  |-  ( `' ( 1st  |`  ( _V  X.  _V ) )
" { x }
)  =  ( { x }  X.  _V )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684   _Vcvv 2788   {csn 3640    X. cxp 4687   `'ccnv 4688    |` cres 4691   "cima 4692    Fn wfn 5250   -->wf 5251   ` cfv 5255   1stc1st 6120   2ndc2nd 6121
This theorem is referenced by:  fparlem3  6220
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-fv 5263  df-1st 6122  df-2nd 6123
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