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Theorem fvelimab 5783
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 2965 . . 3  |-  ( C  e.  ( F " B )  ->  C  e.  _V )
21anim2i 554 . 2  |-  ( ( ( F  Fn  A  /\  B  C_  A )  /\  C  e.  ( F " B ) )  ->  ( ( F  Fn  A  /\  B  C_  A )  /\  C  e.  _V )
)
3 fvex 5743 . . . . 5  |-  ( F `
 x )  e. 
_V
4 eleq1 2497 . . . . 5  |-  ( ( F `  x )  =  C  ->  (
( F `  x
)  e.  _V  <->  C  e.  _V ) )
53, 4mpbii 204 . . . 4  |-  ( ( F `  x )  =  C  ->  C  e.  _V )
65rexlimivw 2827 . . 3  |-  ( E. x  e.  B  ( F `  x )  =  C  ->  C  e.  _V )
76anim2i 554 . 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 2497 . . . . . 6  |-  ( y  =  C  ->  (
y  e.  ( F
" B )  <->  C  e.  ( F " B ) ) )
9 eqeq2 2446 . . . . . . 7  |-  ( y  =  C  ->  (
( F `  x
)  =  y  <->  ( F `  x )  =  C ) )
109rexbidv 2727 . . . . . 6  |-  ( y  =  C  ->  ( E. x  e.  B  ( F `  x )  =  y  <->  E. x  e.  B  ( F `  x )  =  C ) )
118, 10bibi12d 314 . . . . 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 309 . . . 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 5543 . . . . . . 7  |-  ( F  Fn  A  ->  Fun  F )
1413adantr 453 . . . . . 6  |-  ( ( F  Fn  A  /\  B  C_  A )  ->  Fun  F )
15 fndm 5545 . . . . . . . 8  |-  ( F  Fn  A  ->  dom  F  =  A )
1615sseq2d 3377 . . . . . . 7  |-  ( F  Fn  A  ->  ( B  C_  dom  F  <->  B  C_  A
) )
1716biimpar 473 . . . . . 6  |-  ( ( F  Fn  A  /\  B  C_  A )  ->  B  C_  dom  F )
18 dfimafn 5776 . . . . . 6  |-  ( ( Fun  F  /\  B  C_ 
dom  F )  -> 
( F " B
)  =  { y  |  E. x  e.  B  ( F `  x )  =  y } )
1914, 17, 18syl2anc 644 . . . . 5  |-  ( ( F  Fn  A  /\  B  C_  A )  -> 
( F " B
)  =  { y  |  E. x  e.  B  ( F `  x )  =  y } )
2019abeq2d 2546 . . . 4  |-  ( ( F  Fn  A  /\  B  C_  A )  -> 
( y  e.  ( F " B )  <->  E. x  e.  B  ( F `  x )  =  y ) )
2112, 20vtoclg 3012 . . 3  |-  ( C  e.  _V  ->  (
( F  Fn  A  /\  B  C_  A )  ->  ( C  e.  ( F " B
)  <->  E. x  e.  B  ( F `  x )  =  C ) ) )
2221impcom 421 . 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 870 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    <-> wb 178    /\ wa 360    = wceq 1653    e. wcel 1726   {cab 2423   E.wrex 2707   _Vcvv 2957    C_ wss 3321   dom cdm 4879   "cima 4882   Fun wfun 5449    Fn wfn 5450   ` cfv 5455
This theorem is referenced by:  ssimaex  5789  rexima  5978  ralima  5979  f1elima  6010  ovelimab  6225  tcrank  7809  ackbij2  8124  fin1a2lem6  8286  iunfo  8415  grothomex  8705  axpre-sup  9045  injresinjlem  11200  lmhmima  16124  txkgen  17685  fmucndlem  18322  mdegldg  19990  ig1peu  20095  efopn  20550  cusgrares  21482  pjimai  23680  indf1ofs  24424  ballotlemsima  24774  nocvxmin  25647  isnacs2  26761  isnacs3  26765  islmodfg  27145  kercvrlsm  27159  isnumbasgrplem2  27247  dfacbasgrp  27251
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2418  ax-sep 4331  ax-nul 4339  ax-pr 4404
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2286  df-mo 2287  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ne 2602  df-ral 2711  df-rex 2712  df-rab 2715  df-v 2959  df-sbc 3163  df-dif 3324  df-un 3326  df-in 3328  df-ss 3335  df-nul 3630  df-if 3741  df-sn 3821  df-pr 3822  df-op 3824  df-uni 4017  df-br 4214  df-opab 4268  df-id 4499  df-xp 4885  df-rel 4886  df-cnv 4887  df-co 4888  df-dm 4889  df-rn 4890  df-res 4891  df-ima 4892  df-iota 5419  df-fun 5457  df-fn 5458  df-fv 5463
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