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Theorem funimass4f 24049
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 5479 . . . . 5  |-  F/ x Fun  F
3 funimass4f.1 . . . . . 6  |-  F/_ x A
41nfdm 5114 . . . . . 6  |-  F/_ x dom  F
53, 4nfss 3343 . . . . 5  |-  F/ x  A  C_  dom  F
62, 5nfan 1847 . . . 4  |-  F/ x
( Fun  F  /\  A  C_  dom  F )
71, 3nfima 5214 . . . . 5  |-  F/_ x
( F " A
)
8 funimass4f.2 . . . . 5  |-  F/_ x B
97, 8nfss 3343 . . . 4  |-  F/ x
( F " A
)  C_  B
106, 9nfan 1847 . . 3  |-  F/ x
( ( Fun  F  /\  A  C_  dom  F
)  /\  ( F " A )  C_  B
)
11 funfvima2 5977 . . . 4  |-  ( ( Fun  F  /\  A  C_ 
dom  F )  -> 
( x  e.  A  ->  ( F `  x
)  e.  ( F
" A ) ) )
12 ssel 3344 . . . 4  |-  ( ( F " A ) 
C_  B  ->  (
( F `  x
)  e.  ( F
" A )  -> 
( F `  x
)  e.  B ) )
1311, 12sylan9 640 . . 3  |-  ( ( ( Fun  F  /\  A  C_  dom  F )  /\  ( F " A )  C_  B
)  ->  ( x  e.  A  ->  ( F `
 x )  e.  B ) )
1410, 13ralrimi 2789 . 2  |-  ( ( ( Fun  F  /\  A  C_  dom  F )  /\  ( F " A )  C_  B
)  ->  A. x  e.  A  ( F `  x )  e.  B
)
153, 1dfimafnf 24048 . . . 4  |-  ( ( Fun  F  /\  A  C_ 
dom  F )  -> 
( F " A
)  =  { y  |  E. x  e.  A  y  =  ( F `  x ) } )
1615adantr 453 . . 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 23998 . . . 4  |-  ( A. x  e.  A  ( F `  x )  e.  B  ->  { y  |  E. x  e.  A  y  =  ( F `  x ) }  C_  B )
1817adantl 454 . . 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 3384 . 2  |-  ( ( ( Fun  F  /\  A  C_  dom  F )  /\  A. x  e.  A  ( F `  x )  e.  B
)  ->  ( F " A )  C_  B
)
2014, 19impbida 807 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 178    /\ wa 360    = wceq 1653    e. wcel 1726   {cab 2424   F/_wnfc 2561   A.wral 2707   E.wrex 2708    C_ wss 3322   dom cdm 4881   "cima 4884   Fun wfun 5451   ` cfv 5457
This theorem is referenced by:  ballotlem7  24798
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4333  ax-nul 4341  ax-pr 4406
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-sbc 3164  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4216  df-opab 4270  df-id 4501  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-fv 5465
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