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Theorem fvelimab 5344
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 2630 . . . 4  |-  ( C  e.  ( F " B )  ->  C  e.  _V )
21anim2i 334 . . 3  |-  ( ( ( F  Fn  A  /\  B  C_  A )  /\  C  e.  ( F " B ) )  ->  ( ( F  Fn  A  /\  B  C_  A )  /\  C  e.  _V )
)
3 ssel2 3018 . . . . . . . 8  |-  ( ( B  C_  A  /\  u  e.  B )  ->  u  e.  A )
4 funfvex 5306 . . . . . . . . 9  |-  ( ( Fun  F  /\  u  e.  dom  F )  -> 
( F `  u
)  e.  _V )
54funfni 5100 . . . . . . . 8  |-  ( ( F  Fn  A  /\  u  e.  A )  ->  ( F `  u
)  e.  _V )
63, 5sylan2 280 . . . . . . 7  |-  ( ( F  Fn  A  /\  ( B  C_  A  /\  u  e.  B )
)  ->  ( F `  u )  e.  _V )
76anassrs 392 . . . . . 6  |-  ( ( ( F  Fn  A  /\  B  C_  A )  /\  u  e.  B
)  ->  ( F `  u )  e.  _V )
8 eleq1 2150 . . . . . 6  |-  ( ( F `  u )  =  C  ->  (
( F `  u
)  e.  _V  <->  C  e.  _V ) )
97, 8syl5ibcom 153 . . . . 5  |-  ( ( ( F  Fn  A  /\  B  C_  A )  /\  u  e.  B
)  ->  ( ( F `  u )  =  C  ->  C  e. 
_V ) )
109rexlimdva 2489 . . . 4  |-  ( ( F  Fn  A  /\  B  C_  A )  -> 
( E. u  e.  B  ( F `  u )  =  C  ->  C  e.  _V ) )
1110imdistani 434 . . 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 2150 . . . . . . 7  |-  ( v  =  C  ->  (
v  e.  ( F
" B )  <->  C  e.  ( F " B ) ) )
13 eqeq2 2097 . . . . . . . 8  |-  ( v  =  C  ->  (
( F `  u
)  =  v  <->  ( F `  u )  =  C ) )
1413rexbidv 2381 . . . . . . 7  |-  ( v  =  C  ->  ( E. u  e.  B  ( F `  u )  =  v  <->  E. u  e.  B  ( F `  u )  =  C ) )
1512, 14bibi12d 233 . . . . . 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 228 . . . . 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 5097 . . . . . . . 8  |-  ( F  Fn  A  ->  Fun  F )
1817adantr 270 . . . . . . 7  |-  ( ( F  Fn  A  /\  B  C_  A )  ->  Fun  F )
19 fndm 5099 . . . . . . . . 9  |-  ( F  Fn  A  ->  dom  F  =  A )
2019sseq2d 3052 . . . . . . . 8  |-  ( F  Fn  A  ->  ( B  C_  dom  F  <->  B  C_  A
) )
2120biimpar 291 . . . . . . 7  |-  ( ( F  Fn  A  /\  B  C_  A )  ->  B  C_  dom  F )
22 dfimafn 5337 . . . . . . 7  |-  ( ( Fun  F  /\  B  C_ 
dom  F )  -> 
( F " B
)  =  { v  |  E. u  e.  B  ( F `  u )  =  v } )
2318, 21, 22syl2anc 403 . . . . . 6  |-  ( ( F  Fn  A  /\  B  C_  A )  -> 
( F " B
)  =  { v  |  E. u  e.  B  ( F `  u )  =  v } )
2423abeq2d 2200 . . . . 5  |-  ( ( F  Fn  A  /\  B  C_  A )  -> 
( v  e.  ( F " B )  <->  E. u  e.  B  ( F `  u )  =  v ) )
2516, 24vtoclg 2679 . . . 4  |-  ( C  e.  _V  ->  (
( F  Fn  A  /\  B  C_  A )  ->  ( C  e.  ( F " B
)  <->  E. u  e.  B  ( F `  u )  =  C ) ) )
2625impcom 123 . . 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 863 . 2  |-  ( ( F  Fn  A  /\  B  C_  A )  -> 
( C  e.  ( F " B )  <->  E. u  e.  B  ( F `  u )  =  C ) )
28 fveq2 5289 . . . 4  |-  ( u  =  x  ->  ( F `  u )  =  ( F `  x ) )
2928eqeq1d 2096 . . 3  |-  ( u  =  x  ->  (
( F `  u
)  =  C  <->  ( F `  x )  =  C ) )
3029cbvrexv 2591 . 2  |-  ( E. u  e.  B  ( F `  u )  =  C  <->  E. x  e.  B  ( F `  x )  =  C )
3127, 30syl6bb 194 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 102    <-> wb 103    = wceq 1289    e. wcel 1438   {cab 2074   E.wrex 2360   _Vcvv 2619    C_ wss 2997   dom cdm 4428   "cima 4431   Fun wfun 4996    Fn wfn 4997   ` cfv 5002
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3949  ax-pow 4001  ax-pr 4027
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-sbc 2839  df-un 3001  df-in 3003  df-ss 3010  df-pw 3427  df-sn 3447  df-pr 3448  df-op 3450  df-uni 3649  df-br 3838  df-opab 3892  df-id 4111  df-xp 4434  df-rel 4435  df-cnv 4436  df-co 4437  df-dm 4438  df-rn 4439  df-res 4440  df-ima 4441  df-iota 4967  df-fun 5004  df-fn 5005  df-fv 5010
This theorem is referenced by:  ssimaex  5349  foima2  5512  rexima  5516  ralima  5517  f1elima  5534  ovelimab  5777
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