Users' Mathboxes Mathbox for Thierry Arnoux < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  funimass4f Unicode version

Theorem funimass4f 23248
Description: Membership relation for the values of a function whose image is a subclass. (Contributed by Thierry Arnoux, 24-Apr-2017.)
Hypotheses
Ref Expression
funimass4f.1  |-  F/_ x A
funimass4f.2  |-  F/_ x B
funimass4f.3  |-  F/_ x F
Assertion
Ref Expression
funimass4f  |-  ( ( Fun  F  /\  A  C_ 
dom  F )  -> 
( ( F " A )  C_  B  <->  A. x  e.  A  ( F `  x )  e.  B ) )

Proof of Theorem funimass4f
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 funimass4f.3 . . . . . . 7  |-  F/_ x F
21nffun 5359 . . . . . 6  |-  F/ x Fun  F
3 funimass4f.1 . . . . . . 7  |-  F/_ x A
41nfdm 5002 . . . . . . 7  |-  F/_ x dom  F
53, 4nfss 3249 . . . . . 6  |-  F/ x  A  C_  dom  F
62, 5nfan 1829 . . . . 5  |-  F/ x
( Fun  F  /\  A  C_  dom  F )
71, 3nfima 5102 . . . . . 6  |-  F/_ x
( F " A
)
8 funimass4f.2 . . . . . 6  |-  F/_ x B
97, 8nfss 3249 . . . . 5  |-  F/ x
( F " A
)  C_  B
106, 9nfan 1829 . . . 4  |-  F/ x
( ( Fun  F  /\  A  C_  dom  F
)  /\  ( F " A )  C_  B
)
11 funfvima2 5840 . . . . 5  |-  ( ( Fun  F  /\  A  C_ 
dom  F )  -> 
( x  e.  A  ->  ( F `  x
)  e.  ( F
" A ) ) )
12 ssel 3250 . . . . 5  |-  ( ( F " A ) 
C_  B  ->  (
( F `  x
)  e.  ( F
" A )  -> 
( F `  x
)  e.  B ) )
1311, 12sylan9 638 . . . 4  |-  ( ( ( Fun  F  /\  A  C_  dom  F )  /\  ( F " A )  C_  B
)  ->  ( x  e.  A  ->  ( F `
 x )  e.  B ) )
1410, 13ralrimi 2700 . . 3  |-  ( ( ( Fun  F  /\  A  C_  dom  F )  /\  ( F " A )  C_  B
)  ->  A. x  e.  A  ( F `  x )  e.  B
)
1514ex 423 . 2  |-  ( ( Fun  F  /\  A  C_ 
dom  F )  -> 
( ( F " A )  C_  B  ->  A. x  e.  A  ( F `  x )  e.  B ) )
163, 1dfimafnf 23247 . . . . 5  |-  ( ( Fun  F  /\  A  C_ 
dom  F )  -> 
( F " A
)  =  { y  |  E. x  e.  A  y  =  ( F `  x ) } )
1716adantr 451 . . . 4  |-  ( ( ( Fun  F  /\  A  C_  dom  F )  /\  A. x  e.  A  ( F `  x )  e.  B
)  ->  ( F " A )  =  {
y  |  E. x  e.  A  y  =  ( F `  x ) } )
188abrexss 23189 . . . . 5  |-  ( A. x  e.  A  ( F `  x )  e.  B  ->  { y  |  E. x  e.  A  y  =  ( F `  x ) }  C_  B )
1918adantl 452 . . . 4  |-  ( ( ( Fun  F  /\  A  C_  dom  F )  /\  A. x  e.  A  ( F `  x )  e.  B
)  ->  { y  |  E. x  e.  A  y  =  ( F `  x ) }  C_  B )
2017, 19eqsstrd 3288 . . 3  |-  ( ( ( Fun  F  /\  A  C_  dom  F )  /\  A. x  e.  A  ( F `  x )  e.  B
)  ->  ( F " A )  C_  B
)
2120ex 423 . 2  |-  ( ( Fun  F  /\  A  C_ 
dom  F )  -> 
( A. x  e.  A  ( F `  x )  e.  B  ->  ( F " A
)  C_  B )
)
2215, 21impbid 183 1  |-  ( ( Fun  F  /\  A  C_ 
dom  F )  -> 
( ( F " A )  C_  B  <->  A. x  e.  A  ( F `  x )  e.  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1642    e. wcel 1710   {cab 2344   F/_wnfc 2481   A.wral 2619   E.wrex 2620    C_ wss 3228   dom cdm 4771   "cima 4774   Fun wfun 5331   ` cfv 5337
This theorem is referenced by:  ballotlem7  24042
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-sep 4222  ax-nul 4230  ax-pr 4295
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-ral 2624  df-rex 2625  df-rab 2628  df-v 2866  df-sbc 3068  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-nul 3532  df-if 3642  df-sn 3722  df-pr 3723  df-op 3725  df-uni 3909  df-br 4105  df-opab 4159  df-id 4391  df-xp 4777  df-rel 4778  df-cnv 4779  df-co 4780  df-dm 4781  df-rn 4782  df-res 4783  df-ima 4784  df-iota 5301  df-fun 5339  df-fn 5340  df-fv 5345
  Copyright terms: Public domain W3C validator