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Theorem fvresex 5778
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 3211 . . . . . . . 8  |-  A  C_  _V
2 resmpt 5016 . . . . . . . 8  |-  ( A 
C_  _V  ->  ( ( z  e.  _V  |->  ( F `  z ) )  |`  A )  =  ( z  e.  A  |->  ( F `  z ) ) )
31, 2ax-mp 8 . . . . . . 7  |-  ( ( z  e.  _V  |->  ( F `  z ) )  |`  A )  =  ( z  e.  A  |->  ( F `  z ) )
43fveq1i 5542 . . . . . 6  |-  ( ( ( z  e.  _V  |->  ( F `  z ) )  |`  A ) `  x )  =  ( ( z  e.  A  |->  ( F `  z
) ) `  x
)
5 vex 2804 . . . . . . . 8  |-  x  e. 
_V
6 fveq2 5541 . . . . . . . . 9  |-  ( z  =  x  ->  ( F `  z )  =  ( F `  x ) )
7 eqid 2296 . . . . . . . . 9  |-  ( z  e.  _V  |->  ( F `
 z ) )  =  ( z  e. 
_V  |->  ( F `  z ) )
8 fvex 5555 . . . . . . . . 9  |-  ( F `
 x )  e. 
_V
96, 7, 8fvmpt 5618 . . . . . . . 8  |-  ( x  e.  _V  ->  (
( z  e.  _V  |->  ( F `  z ) ) `  x )  =  ( F `  x ) )
105, 9ax-mp 8 . . . . . . 7  |-  ( ( z  e.  _V  |->  ( F `  z ) ) `  x )  =  ( F `  x )
11 fveqres 5576 . . . . . . 7  |-  ( ( ( z  e.  _V  |->  ( F `  z ) ) `  x )  =  ( F `  x )  ->  (
( ( z  e. 
_V  |->  ( F `  z ) )  |`  A ) `  x
)  =  ( ( F  |`  A ) `  x ) )
1210, 11ax-mp 8 . . . . . 6  |-  ( ( ( z  e.  _V  |->  ( F `  z ) )  |`  A ) `  x )  =  ( ( F  |`  A ) `
 x )
134, 12eqtr3i 2318 . . . . 5  |-  ( ( z  e.  A  |->  ( F `  z ) ) `  x )  =  ( ( F  |`  A ) `  x
)
1413eqeq2i 2306 . . . 4  |-  ( y  =  ( ( z  e.  A  |->  ( F `
 z ) ) `
 x )  <->  y  =  ( ( F  |`  A ) `  x
) )
1514exbii 1572 . . 3  |-  ( E. x  y  =  ( ( z  e.  A  |->  ( F `  z
) ) `  x
)  <->  E. x  y  =  ( ( F  |`  A ) `  x
) )
1615abbii 2408 . 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 5762 . . 3  |-  ( z  e.  A  |->  ( F `
 z ) )  e.  _V
1918fvclex 5777 . 2  |-  { y  |  E. x  y  =  ( ( z  e.  A  |->  ( F `
 z ) ) `
 x ) }  e.  _V
2016, 19eqeltrri 2367 1  |-  { y  |  E. x  y  =  ( ( F  |`  A ) `  x
) }  e.  _V
Colors of variables: wff set class
Syntax hints:   E.wex 1531    = wceq 1632    e. wcel 1696   {cab 2282   _Vcvv 2801    C_ wss 3165    e. cmpt 4093    |` cres 4707   ` cfv 5271
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279
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