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Theorem fvelimab 5409
Description: Function value in an image. (Contributed by NM, 20-Jan-2007.) (Proof shortened by Andrew Salmon, 22-Oct-2011.) (Revised by David Abernethy, 17-Dec-2011.)
Assertion
Ref Expression
fvelimab  |-  ( ( F  Fn  A  /\  B  C_  A )  -> 
( C  e.  ( F " B )  <->  E. x  e.  B  ( F `  x )  =  C ) )
Distinct variable groups:    x, B    x, C    x, F
Allowed substitution hint:    A( x)

Proof of Theorem fvelimab
Dummy variables  v  u are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elex 2652 . . . 4  |-  ( C  e.  ( F " B )  ->  C  e.  _V )
21anim2i 337 . . 3  |-  ( ( ( F  Fn  A  /\  B  C_  A )  /\  C  e.  ( F " B ) )  ->  ( ( F  Fn  A  /\  B  C_  A )  /\  C  e.  _V )
)
3 ssel2 3042 . . . . . . . 8  |-  ( ( B  C_  A  /\  u  e.  B )  ->  u  e.  A )
4 funfvex 5370 . . . . . . . . 9  |-  ( ( Fun  F  /\  u  e.  dom  F )  -> 
( F `  u
)  e.  _V )
54funfni 5159 . . . . . . . 8  |-  ( ( F  Fn  A  /\  u  e.  A )  ->  ( F `  u
)  e.  _V )
63, 5sylan2 282 . . . . . . 7  |-  ( ( F  Fn  A  /\  ( B  C_  A  /\  u  e.  B )
)  ->  ( F `  u )  e.  _V )
76anassrs 395 . . . . . 6  |-  ( ( ( F  Fn  A  /\  B  C_  A )  /\  u  e.  B
)  ->  ( F `  u )  e.  _V )
8 eleq1 2162 . . . . . 6  |-  ( ( F `  u )  =  C  ->  (
( F `  u
)  e.  _V  <->  C  e.  _V ) )
97, 8syl5ibcom 154 . . . . 5  |-  ( ( ( F  Fn  A  /\  B  C_  A )  /\  u  e.  B
)  ->  ( ( F `  u )  =  C  ->  C  e. 
_V ) )
109rexlimdva 2508 . . . 4  |-  ( ( F  Fn  A  /\  B  C_  A )  -> 
( E. u  e.  B  ( F `  u )  =  C  ->  C  e.  _V ) )
1110imdistani 437 . . 3  |-  ( ( ( F  Fn  A  /\  B  C_  A )  /\  E. u  e.  B  ( F `  u )  =  C )  ->  ( ( F  Fn  A  /\  B  C_  A )  /\  C  e.  _V )
)
12 eleq1 2162 . . . . . . 7  |-  ( v  =  C  ->  (
v  e.  ( F
" B )  <->  C  e.  ( F " B ) ) )
13 eqeq2 2109 . . . . . . . 8  |-  ( v  =  C  ->  (
( F `  u
)  =  v  <->  ( F `  u )  =  C ) )
1413rexbidv 2397 . . . . . . 7  |-  ( v  =  C  ->  ( E. u  e.  B  ( F `  u )  =  v  <->  E. u  e.  B  ( F `  u )  =  C ) )
1512, 14bibi12d 234 . . . . . 6  |-  ( v  =  C  ->  (
( v  e.  ( F " B )  <->  E. u  e.  B  ( F `  u )  =  v )  <->  ( C  e.  ( F " B
)  <->  E. u  e.  B  ( F `  u )  =  C ) ) )
1615imbi2d 229 . . . . 5  |-  ( v  =  C  ->  (
( ( F  Fn  A  /\  B  C_  A
)  ->  ( v  e.  ( F " B
)  <->  E. u  e.  B  ( F `  u )  =  v ) )  <-> 
( ( F  Fn  A  /\  B  C_  A
)  ->  ( C  e.  ( F " B
)  <->  E. u  e.  B  ( F `  u )  =  C ) ) ) )
17 fnfun 5156 . . . . . . . 8  |-  ( F  Fn  A  ->  Fun  F )
1817adantr 272 . . . . . . 7  |-  ( ( F  Fn  A  /\  B  C_  A )  ->  Fun  F )
19 fndm 5158 . . . . . . . . 9  |-  ( F  Fn  A  ->  dom  F  =  A )
2019sseq2d 3077 . . . . . . . 8  |-  ( F  Fn  A  ->  ( B  C_  dom  F  <->  B  C_  A
) )
2120biimpar 293 . . . . . . 7  |-  ( ( F  Fn  A  /\  B  C_  A )  ->  B  C_  dom  F )
22 dfimafn 5402 . . . . . . 7  |-  ( ( Fun  F  /\  B  C_ 
dom  F )  -> 
( F " B
)  =  { v  |  E. u  e.  B  ( F `  u )  =  v } )
2318, 21, 22syl2anc 406 . . . . . 6  |-  ( ( F  Fn  A  /\  B  C_  A )  -> 
( F " B
)  =  { v  |  E. u  e.  B  ( F `  u )  =  v } )
2423abeq2d 2212 . . . . 5  |-  ( ( F  Fn  A  /\  B  C_  A )  -> 
( v  e.  ( F " B )  <->  E. u  e.  B  ( F `  u )  =  v ) )
2516, 24vtoclg 2701 . . . 4  |-  ( C  e.  _V  ->  (
( F  Fn  A  /\  B  C_  A )  ->  ( C  e.  ( F " B
)  <->  E. u  e.  B  ( F `  u )  =  C ) ) )
2625impcom 124 . . 3  |-  ( ( ( F  Fn  A  /\  B  C_  A )  /\  C  e.  _V )  ->  ( C  e.  ( F " B
)  <->  E. u  e.  B  ( F `  u )  =  C ) )
272, 11, 26pm5.21nd 869 . 2  |-  ( ( F  Fn  A  /\  B  C_  A )  -> 
( C  e.  ( F " B )  <->  E. u  e.  B  ( F `  u )  =  C ) )
28 fveq2 5353 . . . 4  |-  ( u  =  x  ->  ( F `  u )  =  ( F `  x ) )
2928eqeq1d 2108 . . 3  |-  ( u  =  x  ->  (
( F `  u
)  =  C  <->  ( F `  x )  =  C ) )
3029cbvrexv 2613 . 2  |-  ( E. u  e.  B  ( F `  u )  =  C  <->  E. x  e.  B  ( F `  x )  =  C )
3127, 30syl6bb 195 1  |-  ( ( F  Fn  A  /\  B  C_  A )  -> 
( C  e.  ( F " B )  <->  E. x  e.  B  ( F `  x )  =  C ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104    = wceq 1299    e. wcel 1448   {cab 2086   E.wrex 2376   _Vcvv 2641    C_ wss 3021   dom cdm 4477   "cima 4480   Fun wfun 5053    Fn wfn 5054   ` cfv 5059
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-14 1460  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082  ax-sep 3986  ax-pow 4038  ax-pr 4069
This theorem depends on definitions:  df-bi 116  df-3an 932  df-tru 1302  df-nf 1405  df-sb 1704  df-eu 1963  df-mo 1964  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-ral 2380  df-rex 2381  df-v 2643  df-sbc 2863  df-un 3025  df-in 3027  df-ss 3034  df-pw 3459  df-sn 3480  df-pr 3481  df-op 3483  df-uni 3684  df-br 3876  df-opab 3930  df-id 4153  df-xp 4483  df-rel 4484  df-cnv 4485  df-co 4486  df-dm 4487  df-rn 4488  df-res 4489  df-ima 4490  df-iota 5024  df-fun 5061  df-fn 5062  df-fv 5067
This theorem is referenced by:  ssimaex  5414  foima2  5585  rexima  5588  ralima  5589  f1elima  5606  ovelimab  5853
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