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Theorem funimass4f 23042
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 5277 . . . . . 6  |-  F/ x Fun  F
3 funimass4f.1 . . . . . . 7  |-  F/_ x A
41nfdm 4920 . . . . . . 7  |-  F/_ x dom  F
53, 4nfss 3173 . . . . . 6  |-  F/ x  A  C_  dom  F
62, 5nfan 1771 . . . . 5  |-  F/ x
( Fun  F  /\  A  C_  dom  F )
71, 3nfima 5020 . . . . . 6  |-  F/_ x
( F " A
)
8 funimass4f.2 . . . . . 6  |-  F/_ x B
97, 8nfss 3173 . . . . 5  |-  F/ x
( F " A
)  C_  B
106, 9nfan 1771 . . . 4  |-  F/ x
( ( Fun  F  /\  A  C_  dom  F
)  /\  ( F " A )  C_  B
)
11 funfvima2 5754 . . . . 5  |-  ( ( Fun  F  /\  A  C_ 
dom  F )  -> 
( x  e.  A  ->  ( F `  x
)  e.  ( F
" A ) ) )
12 ssel 3174 . . . . 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 2624 . . 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 23041 . . . . 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 23040 . . . . 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 3212 . . 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 1623    e. wcel 1684   {cab 2269   F/_wnfc 2406   A.wral 2543   E.wrex 2544    C_ wss 3152   dom cdm 4689   "cima 4692   Fun wfun 5249   ` cfv 5255
This theorem is referenced by:  ballotlem7  23094
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-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
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-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  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-fv 5263
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