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Theorem fvresex 5762
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 3198 . . . . . . . 8  |-  A  C_  _V
2 resmpt 5000 . . . . . . . 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 5526 . . . . . 6  |-  ( ( ( z  e.  _V  |->  ( F `  z ) )  |`  A ) `  x )  =  ( ( z  e.  A  |->  ( F `  z
) ) `  x
)
5 vex 2791 . . . . . . . 8  |-  x  e. 
_V
6 fveq2 5525 . . . . . . . . 9  |-  ( z  =  x  ->  ( F `  z )  =  ( F `  x ) )
7 eqid 2283 . . . . . . . . 9  |-  ( z  e.  _V  |->  ( F `
 z ) )  =  ( z  e. 
_V  |->  ( F `  z ) )
8 fvex 5539 . . . . . . . . 9  |-  ( F `
 x )  e. 
_V
96, 7, 8fvmpt 5602 . . . . . . . 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 5560 . . . . . . 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 2305 . . . . 5  |-  ( ( z  e.  A  |->  ( F `  z ) ) `  x )  =  ( ( F  |`  A ) `  x
)
1413eqeq2i 2293 . . . 4  |-  ( y  =  ( ( z  e.  A  |->  ( F `
 z ) ) `
 x )  <->  y  =  ( ( F  |`  A ) `  x
) )
1514exbii 1569 . . 3  |-  ( E. x  y  =  ( ( z  e.  A  |->  ( F `  z
) ) `  x
)  <->  E. x  y  =  ( ( F  |`  A ) `  x
) )
1615abbii 2395 . 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 5746 . . 3  |-  ( z  e.  A  |->  ( F `
 z ) )  e.  _V
1918fvclex 5761 . 2  |-  { y  |  E. x  y  =  ( ( z  e.  A  |->  ( F `
 z ) ) `
 x ) }  e.  _V
2016, 19eqeltrri 2354 1  |-  { y  |  E. x  y  =  ( ( F  |`  A ) `  x
) }  e.  _V
Colors of variables: wff set class
Syntax hints:   E.wex 1528    = wceq 1623    e. wcel 1684   {cab 2269   _Vcvv 2788    C_ wss 3152    e. cmpt 4077    |` cres 4691   ` cfv 5255
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-rep 4131  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-reu 2550  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-pw 3627  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-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263
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