MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  fvresex Structured version   Unicode version

Theorem fvresex 6758
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 3509 . . . . . . . 8  |-  A  C_  _V
2 resmpt 5313 . . . . . . . 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 5857 . . . . . 6  |-  ( ( ( z  e.  _V  |->  ( F `  z ) )  |`  A ) `  x )  =  ( ( z  e.  A  |->  ( F `  z
) ) `  x
)
5 vex 3098 . . . . . . . 8  |-  x  e. 
_V
6 fveq2 5856 . . . . . . . . 9  |-  ( z  =  x  ->  ( F `  z )  =  ( F `  x ) )
7 eqid 2443 . . . . . . . . 9  |-  ( z  e.  _V  |->  ( F `
 z ) )  =  ( z  e. 
_V  |->  ( F `  z ) )
8 fvex 5866 . . . . . . . . 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 5890 . . . . . . 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 2474 . . . . 5  |-  ( ( z  e.  A  |->  ( F `  z ) ) `  x )  =  ( ( F  |`  A ) `  x
)
1413eqeq2i 2461 . . . 4  |-  ( y  =  ( ( z  e.  A  |->  ( F `
 z ) ) `
 x )  <->  y  =  ( ( F  |`  A ) `  x
) )
1514exbii 1654 . . 3  |-  ( E. x  y  =  ( ( z  e.  A  |->  ( F `  z
) ) `  x
)  <->  E. x  y  =  ( ( F  |`  A ) `  x
) )
1615abbii 2577 . 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 6128 . . 3  |-  ( z  e.  A  |->  ( F `
 z ) )  e.  _V
1918fvclex 6757 . 2  |-  { y  |  E. x  y  =  ( ( z  e.  A  |->  ( F `
 z ) ) `
 x ) }  e.  _V
2016, 19eqeltrri 2528 1  |-  { y  |  E. x  y  =  ( ( F  |`  A ) `  x
) }  e.  _V
Colors of variables: wff setvar class
Syntax hints:    = wceq 1383   E.wex 1599    e. wcel 1804   {cab 2428   _Vcvv 3095    C_ wss 3461    |-> cmpt 4495    |` cres 4991   ` cfv 5578
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-rep 4548  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676  ax-un 6577
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-ral 2798  df-rex 2799  df-reu 2800  df-rab 2802  df-v 3097  df-sbc 3314  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3771  df-if 3927  df-pw 3999  df-sn 4015  df-pr 4017  df-op 4021  df-uni 4235  df-iun 4317  df-br 4438  df-opab 4496  df-mpt 4497  df-id 4785  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002  df-iota 5541  df-fun 5580  df-fn 5581  df-f 5582  df-f1 5583  df-fo 5584  df-f1o 5585  df-fv 5586
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator