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Theorem funimass4f 23997
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 . . . . . 6  |-  F/_ x F
21nffun 5435 . . . . 5  |-  F/ x Fun  F
3 funimass4f.1 . . . . . 6  |-  F/_ x A
41nfdm 5070 . . . . . 6  |-  F/_ x dom  F
53, 4nfss 3301 . . . . 5  |-  F/ x  A  C_  dom  F
62, 5nfan 1842 . . . 4  |-  F/ x
( Fun  F  /\  A  C_  dom  F )
71, 3nfima 5170 . . . . 5  |-  F/_ x
( F " A
)
8 funimass4f.2 . . . . 5  |-  F/_ x B
97, 8nfss 3301 . . . 4  |-  F/ x
( F " A
)  C_  B
106, 9nfan 1842 . . 3  |-  F/ x
( ( Fun  F  /\  A  C_  dom  F
)  /\  ( F " A )  C_  B
)
11 funfvima2 5933 . . . 4  |-  ( ( Fun  F  /\  A  C_ 
dom  F )  -> 
( x  e.  A  ->  ( F `  x
)  e.  ( F
" A ) ) )
12 ssel 3302 . . . 4  |-  ( ( F " A ) 
C_  B  ->  (
( F `  x
)  e.  ( F
" A )  -> 
( F `  x
)  e.  B ) )
1311, 12sylan9 639 . . 3  |-  ( ( ( Fun  F  /\  A  C_  dom  F )  /\  ( F " A )  C_  B
)  ->  ( x  e.  A  ->  ( F `
 x )  e.  B ) )
1410, 13ralrimi 2747 . 2  |-  ( ( ( Fun  F  /\  A  C_  dom  F )  /\  ( F " A )  C_  B
)  ->  A. x  e.  A  ( F `  x )  e.  B
)
153, 1dfimafnf 23996 . . . 4  |-  ( ( Fun  F  /\  A  C_ 
dom  F )  -> 
( F " A
)  =  { y  |  E. x  e.  A  y  =  ( F `  x ) } )
1615adantr 452 . . 3  |-  ( ( ( Fun  F  /\  A  C_  dom  F )  /\  A. x  e.  A  ( F `  x )  e.  B
)  ->  ( F " A )  =  {
y  |  E. x  e.  A  y  =  ( F `  x ) } )
178abrexss 23946 . . . 4  |-  ( A. x  e.  A  ( F `  x )  e.  B  ->  { y  |  E. x  e.  A  y  =  ( F `  x ) }  C_  B )
1817adantl 453 . . 3  |-  ( ( ( Fun  F  /\  A  C_  dom  F )  /\  A. x  e.  A  ( F `  x )  e.  B
)  ->  { y  |  E. x  e.  A  y  =  ( F `  x ) }  C_  B )
1916, 18eqsstrd 3342 . 2  |-  ( ( ( Fun  F  /\  A  C_  dom  F )  /\  A. x  e.  A  ( F `  x )  e.  B
)  ->  ( F " A )  C_  B
)
2014, 19impbida 806 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 177    /\ wa 359    = wceq 1649    e. wcel 1721   {cab 2390   F/_wnfc 2527   A.wral 2666   E.wrex 2667    C_ wss 3280   dom cdm 4837   "cima 4840   Fun wfun 5407   ` cfv 5413
This theorem is referenced by:  ballotlem7  24746
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pr 4363
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-rab 2675  df-v 2918  df-sbc 3122  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-br 4173  df-opab 4227  df-id 4458  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-fv 5421
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