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Theorem fvresex 6747
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 3517 . . . . . . . 8  |-  A  C_  _V
2 resmpt 5314 . . . . . . . 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 5858 . . . . . 6  |-  ( ( ( z  e.  _V  |->  ( F `  z ) )  |`  A ) `  x )  =  ( ( z  e.  A  |->  ( F `  z
) ) `  x
)
5 vex 3109 . . . . . . . 8  |-  x  e. 
_V
6 fveq2 5857 . . . . . . . . 9  |-  ( z  =  x  ->  ( F `  z )  =  ( F `  x ) )
7 eqid 2460 . . . . . . . . 9  |-  ( z  e.  _V  |->  ( F `
 z ) )  =  ( z  e. 
_V  |->  ( F `  z ) )
8 fvex 5867 . . . . . . . . 9  |-  ( F `
 x )  e. 
_V
96, 7, 8fvmpt 5941 . . . . . . . 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 5891 . . . . . . 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 2491 . . . . 5  |-  ( ( z  e.  A  |->  ( F `  z ) ) `  x )  =  ( ( F  |`  A ) `  x
)
1413eqeq2i 2478 . . . 4  |-  ( y  =  ( ( z  e.  A  |->  ( F `
 z ) ) `
 x )  <->  y  =  ( ( F  |`  A ) `  x
) )
1514exbii 1639 . . 3  |-  ( E. x  y  =  ( ( z  e.  A  |->  ( F `  z
) ) `  x
)  <->  E. x  y  =  ( ( F  |`  A ) `  x
) )
1615abbii 2594 . 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 6122 . . 3  |-  ( z  e.  A  |->  ( F `
 z ) )  e.  _V
1918fvclex 6746 . 2  |-  { y  |  E. x  y  =  ( ( z  e.  A  |->  ( F `
 z ) ) `
 x ) }  e.  _V
2016, 19eqeltrri 2545 1  |-  { y  |  E. x  y  =  ( ( F  |`  A ) `  x
) }  e.  _V
Colors of variables: wff setvar class
Syntax hints:    = wceq 1374   E.wex 1591    e. wcel 1762   {cab 2445   _Vcvv 3106    C_ wss 3469    |-> cmpt 4498    |` cres 4994   ` cfv 5579
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-rep 4551  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-op 4027  df-uni 4239  df-iun 4320  df-br 4441  df-opab 4499  df-mpt 4500  df-id 4788  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587
This theorem is referenced by: (None)
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