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Theorem fvresex 6652
Description: Existence of the class of values of a restricted class. (Contributed by NM, 14-Nov-1995.) (Revised by Mario Carneiro, 11-Sep-2015.)
Hypothesis
Ref Expression
fvresex.1  |-  A  e. 
_V
Assertion
Ref Expression
fvresex  |-  { y  |  E. x  y  =  ( ( F  |`  A ) `  x
) }  e.  _V
Distinct variable groups:    x, y, A    x, F, y

Proof of Theorem fvresex
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 ssv 3476 . . . . . . . 8  |-  A  C_  _V
2 resmpt 5256 . . . . . . . 8  |-  ( A 
C_  _V  ->  ( ( z  e.  _V  |->  ( F `  z ) )  |`  A )  =  ( z  e.  A  |->  ( F `  z ) ) )
31, 2ax-mp 5 . . . . . . 7  |-  ( ( z  e.  _V  |->  ( F `  z ) )  |`  A )  =  ( z  e.  A  |->  ( F `  z ) )
43fveq1i 5792 . . . . . 6  |-  ( ( ( z  e.  _V  |->  ( F `  z ) )  |`  A ) `  x )  =  ( ( z  e.  A  |->  ( F `  z
) ) `  x
)
5 vex 3073 . . . . . . . 8  |-  x  e. 
_V
6 fveq2 5791 . . . . . . . . 9  |-  ( z  =  x  ->  ( F `  z )  =  ( F `  x ) )
7 eqid 2451 . . . . . . . . 9  |-  ( z  e.  _V  |->  ( F `
 z ) )  =  ( z  e. 
_V  |->  ( F `  z ) )
8 fvex 5801 . . . . . . . . 9  |-  ( F `
 x )  e. 
_V
96, 7, 8fvmpt 5875 . . . . . . . 8  |-  ( x  e.  _V  ->  (
( z  e.  _V  |->  ( F `  z ) ) `  x )  =  ( F `  x ) )
105, 9ax-mp 5 . . . . . . 7  |-  ( ( z  e.  _V  |->  ( F `  z ) ) `  x )  =  ( F `  x )
11 fveqres 5825 . . . . . . 7  |-  ( ( ( z  e.  _V  |->  ( F `  z ) ) `  x )  =  ( F `  x )  ->  (
( ( z  e. 
_V  |->  ( F `  z ) )  |`  A ) `  x
)  =  ( ( F  |`  A ) `  x ) )
1210, 11ax-mp 5 . . . . . 6  |-  ( ( ( z  e.  _V  |->  ( F `  z ) )  |`  A ) `  x )  =  ( ( F  |`  A ) `
 x )
134, 12eqtr3i 2482 . . . . 5  |-  ( ( z  e.  A  |->  ( F `  z ) ) `  x )  =  ( ( F  |`  A ) `  x
)
1413eqeq2i 2469 . . . 4  |-  ( y  =  ( ( z  e.  A  |->  ( F `
 z ) ) `
 x )  <->  y  =  ( ( F  |`  A ) `  x
) )
1514exbii 1635 . . 3  |-  ( E. x  y  =  ( ( z  e.  A  |->  ( F `  z
) ) `  x
)  <->  E. x  y  =  ( ( F  |`  A ) `  x
) )
1615abbii 2585 . 2  |-  { y  |  E. x  y  =  ( ( z  e.  A  |->  ( F `
 z ) ) `
 x ) }  =  { y  |  E. x  y  =  ( ( F  |`  A ) `  x
) }
17 fvresex.1 . . . 4  |-  A  e. 
_V
1817mptex 6049 . . 3  |-  ( z  e.  A  |->  ( F `
 z ) )  e.  _V
1918fvclex 6651 . 2  |-  { y  |  E. x  y  =  ( ( z  e.  A  |->  ( F `
 z ) ) `
 x ) }  e.  _V
2016, 19eqeltrri 2536 1  |-  { y  |  E. x  y  =  ( ( F  |`  A ) `  x
) }  e.  _V
Colors of variables: wff setvar class
Syntax hints:    = wceq 1370   E.wex 1587    e. wcel 1758   {cab 2436   _Vcvv 3070    C_ wss 3428    |-> cmpt 4450    |` cres 4942   ` cfv 5518
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-rep 4503  ax-sep 4513  ax-nul 4521  ax-pow 4570  ax-pr 4631  ax-un 6474
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-reu 2802  df-rab 2804  df-v 3072  df-sbc 3287  df-csb 3389  df-dif 3431  df-un 3433  df-in 3435  df-ss 3442  df-nul 3738  df-if 3892  df-pw 3962  df-sn 3978  df-pr 3980  df-op 3984  df-uni 4192  df-iun 4273  df-br 4393  df-opab 4451  df-mpt 4452  df-id 4736  df-xp 4946  df-rel 4947  df-cnv 4948  df-co 4949  df-dm 4950  df-rn 4951  df-res 4952  df-ima 4953  df-iota 5481  df-fun 5520  df-fn 5521  df-f 5522  df-f1 5523  df-fo 5524  df-f1o 5525  df-fv 5526
This theorem is referenced by: (None)
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