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Theorem fvelimab 5585
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 2760 . . . 4  |-  ( C  e.  ( F " B )  ->  C  e.  _V )
21anim2i 342 . . 3  |-  ( ( ( F  Fn  A  /\  B  C_  A )  /\  C  e.  ( F " B ) )  ->  ( ( F  Fn  A  /\  B  C_  A )  /\  C  e.  _V )
)
3 ssel2 3162 . . . . . . . 8  |-  ( ( B  C_  A  /\  u  e.  B )  ->  u  e.  A )
4 funfvex 5544 . . . . . . . . 9  |-  ( ( Fun  F  /\  u  e.  dom  F )  -> 
( F `  u
)  e.  _V )
54funfni 5328 . . . . . . . 8  |-  ( ( F  Fn  A  /\  u  e.  A )  ->  ( F `  u
)  e.  _V )
63, 5sylan2 286 . . . . . . 7  |-  ( ( F  Fn  A  /\  ( B  C_  A  /\  u  e.  B )
)  ->  ( F `  u )  e.  _V )
76anassrs 400 . . . . . 6  |-  ( ( ( F  Fn  A  /\  B  C_  A )  /\  u  e.  B
)  ->  ( F `  u )  e.  _V )
8 eleq1 2250 . . . . . 6  |-  ( ( F `  u )  =  C  ->  (
( F `  u
)  e.  _V  <->  C  e.  _V ) )
97, 8syl5ibcom 155 . . . . 5  |-  ( ( ( F  Fn  A  /\  B  C_  A )  /\  u  e.  B
)  ->  ( ( F `  u )  =  C  ->  C  e. 
_V ) )
109rexlimdva 2604 . . . 4  |-  ( ( F  Fn  A  /\  B  C_  A )  -> 
( E. u  e.  B  ( F `  u )  =  C  ->  C  e.  _V ) )
1110imdistani 445 . . 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 2250 . . . . . . 7  |-  ( v  =  C  ->  (
v  e.  ( F
" B )  <->  C  e.  ( F " B ) ) )
13 eqeq2 2197 . . . . . . . 8  |-  ( v  =  C  ->  (
( F `  u
)  =  v  <->  ( F `  u )  =  C ) )
1413rexbidv 2488 . . . . . . 7  |-  ( v  =  C  ->  ( E. u  e.  B  ( F `  u )  =  v  <->  E. u  e.  B  ( F `  u )  =  C ) )
1512, 14bibi12d 235 . . . . . 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 230 . . . . 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 5325 . . . . . . . 8  |-  ( F  Fn  A  ->  Fun  F )
1817adantr 276 . . . . . . 7  |-  ( ( F  Fn  A  /\  B  C_  A )  ->  Fun  F )
19 fndm 5327 . . . . . . . . 9  |-  ( F  Fn  A  ->  dom  F  =  A )
2019sseq2d 3197 . . . . . . . 8  |-  ( F  Fn  A  ->  ( B  C_  dom  F  <->  B  C_  A
) )
2120biimpar 297 . . . . . . 7  |-  ( ( F  Fn  A  /\  B  C_  A )  ->  B  C_  dom  F )
22 dfimafn 5577 . . . . . . 7  |-  ( ( Fun  F  /\  B  C_ 
dom  F )  -> 
( F " B
)  =  { v  |  E. u  e.  B  ( F `  u )  =  v } )
2318, 21, 22syl2anc 411 . . . . . 6  |-  ( ( F  Fn  A  /\  B  C_  A )  -> 
( F " B
)  =  { v  |  E. u  e.  B  ( F `  u )  =  v } )
2423abeq2d 2300 . . . . 5  |-  ( ( F  Fn  A  /\  B  C_  A )  -> 
( v  e.  ( F " B )  <->  E. u  e.  B  ( F `  u )  =  v ) )
2516, 24vtoclg 2809 . . . 4  |-  ( C  e.  _V  ->  (
( F  Fn  A  /\  B  C_  A )  ->  ( C  e.  ( F " B
)  <->  E. u  e.  B  ( F `  u )  =  C ) ) )
2625impcom 125 . . 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 917 . 2  |-  ( ( F  Fn  A  /\  B  C_  A )  -> 
( C  e.  ( F " B )  <->  E. u  e.  B  ( F `  u )  =  C ) )
28 fveq2 5527 . . . 4  |-  ( u  =  x  ->  ( F `  u )  =  ( F `  x ) )
2928eqeq1d 2196 . . 3  |-  ( u  =  x  ->  (
( F `  u
)  =  C  <->  ( F `  x )  =  C ) )
3029cbvrexv 2716 . 2  |-  ( E. u  e.  B  ( F `  u )  =  C  <->  E. x  e.  B  ( F `  x )  =  C )
3127, 30bitrdi 196 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 104    <-> wb 105    = wceq 1363    e. wcel 2158   {cab 2173   E.wrex 2466   _Vcvv 2749    C_ wss 3141   dom cdm 4638   "cima 4641   Fun wfun 5222    Fn wfn 5223   ` cfv 5228
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-14 2161  ax-ext 2169  ax-sep 4133  ax-pow 4186  ax-pr 4221
This theorem depends on definitions:  df-bi 117  df-3an 981  df-tru 1366  df-nf 1471  df-sb 1773  df-eu 2039  df-mo 2040  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ral 2470  df-rex 2471  df-v 2751  df-sbc 2975  df-un 3145  df-in 3147  df-ss 3154  df-pw 3589  df-sn 3610  df-pr 3611  df-op 3613  df-uni 3822  df-br 4016  df-opab 4077  df-id 4305  df-xp 4644  df-rel 4645  df-cnv 4646  df-co 4647  df-dm 4648  df-rn 4649  df-res 4650  df-ima 4651  df-iota 5190  df-fun 5230  df-fn 5231  df-fv 5236
This theorem is referenced by:  ssimaex  5590  foima2  5765  rexima  5768  ralima  5769  f1elima  5787  ovelimab  6038
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