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Theorem fparlem1 6305
Description: Lemma for fpar 6309. (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 5625 . . . . . 6  |-  ( y  e.  ( _V  X.  _V )  ->  ( ( 1st  |`  ( _V  X.  _V ) ) `  y )  =  ( 1st `  y ) )
21eqeq1d 2366 . . . . 5  |-  ( y  e.  ( _V  X.  _V )  ->  ( ( ( 1st  |`  ( _V  X.  _V ) ) `
 y )  =  x  <->  ( 1st `  y
)  =  x ) )
3 vex 2867 . . . . . . 7  |-  x  e. 
_V
43elsnc2 3745 . . . . . 6  |-  ( ( 1st `  y )  e.  { x }  <->  ( 1st `  y )  =  x )
5 fvex 5622 . . . . . . 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 6228 . . . 4  |-  ( 1st  |`  ( _V  X.  _V ) ) : ( _V  X.  _V ) --> _V
11 ffn 5472 . . . 4  |-  ( ( 1st  |`  ( _V  X.  _V ) ) : ( _V  X.  _V )
--> _V  ->  ( 1st  |`  ( _V  X.  _V ) )  Fn  ( _V  X.  _V ) )
12 fniniseg 5729 . . . 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 6239 . . 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 2355 1  |-  ( `' ( 1st  |`  ( _V  X.  _V ) )
" { x }
)  =  ( { x }  X.  _V )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    /\ wa 358    = wceq 1642    e. wcel 1710   _Vcvv 2864   {csn 3716    X. cxp 4769   `'ccnv 4770    |` cres 4773   "cima 4774    Fn wfn 5332   -->wf 5333   ` cfv 5337   1stc1st 6207   2ndc2nd 6208
This theorem is referenced by:  fparlem3  6307
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-sep 4222  ax-nul 4230  ax-pow 4269  ax-pr 4295  ax-un 4594
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-ral 2624  df-rex 2625  df-rab 2628  df-v 2866  df-sbc 3068  df-csb 3158  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-nul 3532  df-if 3642  df-sn 3722  df-pr 3723  df-op 3725  df-uni 3909  df-iun 3988  df-br 4105  df-opab 4159  df-mpt 4160  df-id 4391  df-xp 4777  df-rel 4778  df-cnv 4779  df-co 4780  df-dm 4781  df-rn 4782  df-res 4783  df-ima 4784  df-iota 5301  df-fun 5339  df-fn 5340  df-f 5341  df-fv 5345  df-1st 6209  df-2nd 6210
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