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Theorem fvelimab 5861
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 variable  y is distinct from all other variables.
StepHypRef Expression
1 elex 3067 . . 3  |-  ( C  e.  ( F " B )  ->  C  e.  _V )
21anim2i 567 . 2  |-  ( ( ( F  Fn  A  /\  B  C_  A )  /\  C  e.  ( F " B ) )  ->  ( ( F  Fn  A  /\  B  C_  A )  /\  C  e.  _V )
)
3 fvex 5815 . . . . 5  |-  ( F `
 x )  e. 
_V
4 eleq1 2474 . . . . 5  |-  ( ( F `  x )  =  C  ->  (
( F `  x
)  e.  _V  <->  C  e.  _V ) )
53, 4mpbii 211 . . . 4  |-  ( ( F `  x )  =  C  ->  C  e.  _V )
65rexlimivw 2892 . . 3  |-  ( E. x  e.  B  ( F `  x )  =  C  ->  C  e.  _V )
76anim2i 567 . 2  |-  ( ( ( F  Fn  A  /\  B  C_  A )  /\  E. x  e.  B  ( F `  x )  =  C )  ->  ( ( F  Fn  A  /\  B  C_  A )  /\  C  e.  _V )
)
8 eleq1 2474 . . . . . 6  |-  ( y  =  C  ->  (
y  e.  ( F
" B )  <->  C  e.  ( F " B ) ) )
9 eqeq2 2417 . . . . . . 7  |-  ( y  =  C  ->  (
( F `  x
)  =  y  <->  ( F `  x )  =  C ) )
109rexbidv 2917 . . . . . 6  |-  ( y  =  C  ->  ( E. x  e.  B  ( F `  x )  =  y  <->  E. x  e.  B  ( F `  x )  =  C ) )
118, 10bibi12d 319 . . . . 5  |-  ( y  =  C  ->  (
( y  e.  ( F " B )  <->  E. x  e.  B  ( F `  x )  =  y )  <->  ( C  e.  ( F " B
)  <->  E. x  e.  B  ( F `  x )  =  C ) ) )
1211imbi2d 314 . . . 4  |-  ( y  =  C  ->  (
( ( F  Fn  A  /\  B  C_  A
)  ->  ( y  e.  ( F " B
)  <->  E. x  e.  B  ( F `  x )  =  y ) )  <-> 
( ( F  Fn  A  /\  B  C_  A
)  ->  ( C  e.  ( F " B
)  <->  E. x  e.  B  ( F `  x )  =  C ) ) ) )
13 fnfun 5615 . . . . . . 7  |-  ( F  Fn  A  ->  Fun  F )
1413adantr 463 . . . . . 6  |-  ( ( F  Fn  A  /\  B  C_  A )  ->  Fun  F )
15 fndm 5617 . . . . . . . 8  |-  ( F  Fn  A  ->  dom  F  =  A )
1615sseq2d 3469 . . . . . . 7  |-  ( F  Fn  A  ->  ( B  C_  dom  F  <->  B  C_  A
) )
1716biimpar 483 . . . . . 6  |-  ( ( F  Fn  A  /\  B  C_  A )  ->  B  C_  dom  F )
18 dfimafn 5854 . . . . . 6  |-  ( ( Fun  F  /\  B  C_ 
dom  F )  -> 
( F " B
)  =  { y  |  E. x  e.  B  ( F `  x )  =  y } )
1914, 17, 18syl2anc 659 . . . . 5  |-  ( ( F  Fn  A  /\  B  C_  A )  -> 
( F " B
)  =  { y  |  E. x  e.  B  ( F `  x )  =  y } )
2019abeq2d 2528 . . . 4  |-  ( ( F  Fn  A  /\  B  C_  A )  -> 
( y  e.  ( F " B )  <->  E. x  e.  B  ( F `  x )  =  y ) )
2112, 20vtoclg 3116 . . 3  |-  ( C  e.  _V  ->  (
( F  Fn  A  /\  B  C_  A )  ->  ( C  e.  ( F " B
)  <->  E. x  e.  B  ( F `  x )  =  C ) ) )
2221impcom 428 . 2  |-  ( ( ( F  Fn  A  /\  B  C_  A )  /\  C  e.  _V )  ->  ( C  e.  ( F " B
)  <->  E. x  e.  B  ( F `  x )  =  C ) )
232, 7, 22pm5.21nd 901 1  |-  ( ( F  Fn  A  /\  B  C_  A )  -> 
( C  e.  ( F " B )  <->  E. x  e.  B  ( F `  x )  =  C ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367    = wceq 1405    e. wcel 1842   {cab 2387   E.wrex 2754   _Vcvv 3058    C_ wss 3413   dom cdm 4942   "cima 4945   Fun wfun 5519    Fn wfn 5520   ` cfv 5525
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-sep 4516  ax-nul 4524  ax-pr 4629
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2758  df-rex 2759  df-rab 2762  df-v 3060  df-sbc 3277  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-nul 3738  df-if 3885  df-sn 3972  df-pr 3974  df-op 3978  df-uni 4191  df-br 4395  df-opab 4453  df-id 4737  df-xp 4948  df-rel 4949  df-cnv 4950  df-co 4951  df-dm 4952  df-rn 4953  df-res 4954  df-ima 4955  df-iota 5489  df-fun 5527  df-fn 5528  df-fv 5533
This theorem is referenced by:  ssimaex  5870  rexima  6088  ralima  6089  f1elima  6108  ovelimab  6390  tcrank  8254  ackbij2  8575  fin1a2lem6  8737  iunfo  8866  grothomex  9157  axpre-sup  9496  injresinjlem  11875  lmhmima  17905  txkgen  20337  fmucndlem  20978  mdegldg  22650  ig1peu  22756  efopn  23225  cusgrares  24770  pjimai  27388  fimarab  27806  fimaproj  28169  qtophaus  28172  indf1ofs  28353  eulerpartgbij  28697  eulerpartlemgvv  28701  ballotlemsima  28840  elmthm  29669  nocvxmin  30124  isnacs2  34981  isnacs3  34985  islmodfg  35358  kercvrlsm  35372  isnumbasgrplem2  35398  dfacbasgrp  35402  fvelimabd  35972  unima  36797  fourierdlem62  37301
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