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Theorem fvelimab 5768
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 3002 . . 3  |-  ( C  e.  ( F " B )  ->  C  e.  _V )
21anim2i 569 . 2  |-  ( ( ( F  Fn  A  /\  B  C_  A )  /\  C  e.  ( F " B ) )  ->  ( ( F  Fn  A  /\  B  C_  A )  /\  C  e.  _V )
)
3 fvex 5722 . . . . 5  |-  ( F `
 x )  e. 
_V
4 eleq1 2503 . . . . 5  |-  ( ( F `  x )  =  C  ->  (
( F `  x
)  e.  _V  <->  C  e.  _V ) )
53, 4mpbii 211 . . . 4  |-  ( ( F `  x )  =  C  ->  C  e.  _V )
65rexlimivw 2858 . . 3  |-  ( E. x  e.  B  ( F `  x )  =  C  ->  C  e.  _V )
76anim2i 569 . 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 2503 . . . . . 6  |-  ( y  =  C  ->  (
y  e.  ( F
" B )  <->  C  e.  ( F " B ) ) )
9 eqeq2 2452 . . . . . . 7  |-  ( y  =  C  ->  (
( F `  x
)  =  y  <->  ( F `  x )  =  C ) )
109rexbidv 2757 . . . . . 6  |-  ( y  =  C  ->  ( E. x  e.  B  ( F `  x )  =  y  <->  E. x  e.  B  ( F `  x )  =  C ) )
118, 10bibi12d 321 . . . . 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 316 . . . 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 5529 . . . . . . 7  |-  ( F  Fn  A  ->  Fun  F )
1413adantr 465 . . . . . 6  |-  ( ( F  Fn  A  /\  B  C_  A )  ->  Fun  F )
15 fndm 5531 . . . . . . . 8  |-  ( F  Fn  A  ->  dom  F  =  A )
1615sseq2d 3405 . . . . . . 7  |-  ( F  Fn  A  ->  ( B  C_  dom  F  <->  B  C_  A
) )
1716biimpar 485 . . . . . 6  |-  ( ( F  Fn  A  /\  B  C_  A )  ->  B  C_  dom  F )
18 dfimafn 5761 . . . . . 6  |-  ( ( Fun  F  /\  B  C_ 
dom  F )  -> 
( F " B
)  =  { y  |  E. x  e.  B  ( F `  x )  =  y } )
1914, 17, 18syl2anc 661 . . . . 5  |-  ( ( F  Fn  A  /\  B  C_  A )  -> 
( F " B
)  =  { y  |  E. x  e.  B  ( F `  x )  =  y } )
2019abeq2d 2553 . . . 4  |-  ( ( F  Fn  A  /\  B  C_  A )  -> 
( y  e.  ( F " B )  <->  E. x  e.  B  ( F `  x )  =  y ) )
2112, 20vtoclg 3051 . . 3  |-  ( C  e.  _V  ->  (
( F  Fn  A  /\  B  C_  A )  ->  ( C  e.  ( F " B
)  <->  E. x  e.  B  ( F `  x )  =  C ) ) )
2221impcom 430 . 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 893 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 369    = wceq 1369    e. wcel 1756   {cab 2429   E.wrex 2737   _Vcvv 2993    C_ wss 3349   dom cdm 4861   "cima 4864   Fun wfun 5433    Fn wfn 5434   ` cfv 5439
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4434  ax-nul 4442  ax-pr 4552
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2622  df-ral 2741  df-rex 2742  df-rab 2745  df-v 2995  df-sbc 3208  df-dif 3352  df-un 3354  df-in 3356  df-ss 3363  df-nul 3659  df-if 3813  df-sn 3899  df-pr 3901  df-op 3905  df-uni 4113  df-br 4314  df-opab 4372  df-id 4657  df-xp 4867  df-rel 4868  df-cnv 4869  df-co 4870  df-dm 4871  df-rn 4872  df-res 4873  df-ima 4874  df-iota 5402  df-fun 5441  df-fn 5442  df-fv 5447
This theorem is referenced by:  ssimaex  5777  rexima  5977  ralima  5978  f1elima  5997  ovelimab  6262  tcrank  8112  ackbij2  8433  fin1a2lem6  8595  iunfo  8724  grothomex  9017  axpre-sup  9357  injresinjlem  11659  lmhmima  17150  txkgen  19247  fmucndlem  19888  mdegldg  21559  ig1peu  21665  efopn  22125  cusgrares  23402  pjimai  25602  indf1ofs  26504  eulerpartgbij  26777  eulerpartlemgvv  26781  ballotlemsima  26920  nocvxmin  27854  isnacs2  29068  isnacs3  29072  islmodfg  29448  kercvrlsm  29462  isnumbasgrplem2  29486  dfacbasgrp  29490
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