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Theorem fparlem1 6775
Description: Lemma for fpar 6779. (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 5806 . . . . . 6  |-  ( y  e.  ( _V  X.  _V )  ->  ( ( 1st  |`  ( _V  X.  _V ) ) `  y )  =  ( 1st `  y ) )
21eqeq1d 2453 . . . . 5  |-  ( y  e.  ( _V  X.  _V )  ->  ( ( ( 1st  |`  ( _V  X.  _V ) ) `
 y )  =  x  <->  ( 1st `  y
)  =  x ) )
3 vex 3074 . . . . . . 7  |-  x  e. 
_V
43elsnc2 4009 . . . . . 6  |-  ( ( 1st `  y )  e.  { x }  <->  ( 1st `  y )  =  x )
5 fvex 5802 . . . . . . 7  |-  ( 2nd `  y )  e.  _V
65biantru 505 . . . . . 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 637 . . 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 6701 . . . 4  |-  ( 1st  |`  ( _V  X.  _V ) ) : ( _V  X.  _V ) --> _V
11 ffn 5660 . . . 4  |-  ( ( 1st  |`  ( _V  X.  _V ) ) : ( _V  X.  _V )
--> _V  ->  ( 1st  |`  ( _V  X.  _V ) )  Fn  ( _V  X.  _V ) )
12 fniniseg 5926 . . . 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 6712 . . 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 2447 1  |-  ( `' ( 1st  |`  ( _V  X.  _V ) )
" { x }
)  =  ( { x }  X.  _V )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    /\ wa 369    = wceq 1370    e. wcel 1758   _Vcvv 3071   {csn 3978    X. cxp 4939   `'ccnv 4940    |` cres 4943   "cima 4944    Fn wfn 5514   -->wf 5515   ` cfv 5519   1stc1st 6678   2ndc2nd 6679
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-sep 4514  ax-nul 4522  ax-pow 4571  ax-pr 4632  ax-un 6475
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-rab 2804  df-v 3073  df-sbc 3288  df-csb 3390  df-dif 3432  df-un 3434  df-in 3436  df-ss 3443  df-nul 3739  df-if 3893  df-sn 3979  df-pr 3981  df-op 3985  df-uni 4193  df-iun 4274  df-br 4394  df-opab 4452  df-mpt 4453  df-id 4737  df-xp 4947  df-rel 4948  df-cnv 4949  df-co 4950  df-dm 4951  df-rn 4952  df-res 4953  df-ima 4954  df-iota 5482  df-fun 5521  df-fn 5522  df-f 5523  df-fv 5527  df-1st 6680  df-2nd 6681
This theorem is referenced by:  fparlem3  6777
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