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Theorem fvelimab 5616
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 2830 . . 3  |-  ( C  e.  ( F " B )  ->  C  e.  _V )
21anim2i 552 . 2  |-  ( ( ( F  Fn  A  /\  B  C_  A )  /\  C  e.  ( F " B ) )  ->  ( ( F  Fn  A  /\  B  C_  A )  /\  C  e.  _V )
)
3 fvex 5577 . . . . 5  |-  ( F `
 x )  e. 
_V
4 eleq1 2376 . . . . 5  |-  ( ( F `  x )  =  C  ->  (
( F `  x
)  e.  _V  <->  C  e.  _V ) )
53, 4mpbii 202 . . . 4  |-  ( ( F `  x )  =  C  ->  C  e.  _V )
65rexlimivw 2697 . . 3  |-  ( E. x  e.  B  ( F `  x )  =  C  ->  C  e.  _V )
76anim2i 552 . 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 2376 . . . . . 6  |-  ( y  =  C  ->  (
y  e.  ( F
" B )  <->  C  e.  ( F " B ) ) )
9 eqeq2 2325 . . . . . . 7  |-  ( y  =  C  ->  (
( F `  x
)  =  y  <->  ( F `  x )  =  C ) )
109rexbidv 2598 . . . . . 6  |-  ( y  =  C  ->  ( E. x  e.  B  ( F `  x )  =  y  <->  E. x  e.  B  ( F `  x )  =  C ) )
118, 10bibi12d 312 . . . . 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 307 . . . 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 5378 . . . . . . 7  |-  ( F  Fn  A  ->  Fun  F )
1413adantr 451 . . . . . 6  |-  ( ( F  Fn  A  /\  B  C_  A )  ->  Fun  F )
15 fndm 5380 . . . . . . . 8  |-  ( F  Fn  A  ->  dom  F  =  A )
1615sseq2d 3240 . . . . . . 7  |-  ( F  Fn  A  ->  ( B  C_  dom  F  <->  B  C_  A
) )
1716biimpar 471 . . . . . 6  |-  ( ( F  Fn  A  /\  B  C_  A )  ->  B  C_  dom  F )
18 dfimafn 5609 . . . . . 6  |-  ( ( Fun  F  /\  B  C_ 
dom  F )  -> 
( F " B
)  =  { y  |  E. x  e.  B  ( F `  x )  =  y } )
1914, 17, 18syl2anc 642 . . . . 5  |-  ( ( F  Fn  A  /\  B  C_  A )  -> 
( F " B
)  =  { y  |  E. x  e.  B  ( F `  x )  =  y } )
2019abeq2d 2425 . . . 4  |-  ( ( F  Fn  A  /\  B  C_  A )  -> 
( y  e.  ( F " B )  <->  E. x  e.  B  ( F `  x )  =  y ) )
2112, 20vtoclg 2877 . . 3  |-  ( C  e.  _V  ->  (
( F  Fn  A  /\  B  C_  A )  ->  ( C  e.  ( F " B
)  <->  E. x  e.  B  ( F `  x )  =  C ) ) )
2221impcom 419 . 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 868 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 176    /\ wa 358    = wceq 1633    e. wcel 1701   {cab 2302   E.wrex 2578   _Vcvv 2822    C_ wss 3186   dom cdm 4726   "cima 4729   Fun wfun 5286    Fn wfn 5287   ` cfv 5292
This theorem is referenced by:  ssimaex  5622  rexima  5798  ralima  5799  f1elima  5829  ovelimab  6040  tcrank  7599  ackbij2  7914  fin1a2lem6  8076  iunfo  8206  grothomex  8496  axpre-sup  8836  lmhmima  15853  txkgen  17402  mdegldg  19505  ig1peu  19610  efopn  20058  pjimai  22811  fmucndlem  23483  indf1ofs  23838  ballotlemsima  23947  nocvxmin  24730  isnacs2  25929  isnacs3  25933  islmodfg  26315  kercvrlsm  26329  isnumbasgrplem2  26417  dfacbasgrp  26421  injresinjlem  27276
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-14 1705  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897  ax-ext 2297  ax-sep 4178  ax-nul 4186  ax-pr 4251
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1533  df-nf 1536  df-sb 1640  df-eu 2180  df-mo 2181  df-clab 2303  df-cleq 2309  df-clel 2312  df-nfc 2441  df-ne 2481  df-ral 2582  df-rex 2583  df-rab 2586  df-v 2824  df-sbc 3026  df-dif 3189  df-un 3191  df-in 3193  df-ss 3200  df-nul 3490  df-if 3600  df-sn 3680  df-pr 3681  df-op 3683  df-uni 3865  df-br 4061  df-opab 4115  df-id 4346  df-xp 4732  df-rel 4733  df-cnv 4734  df-co 4735  df-dm 4736  df-rn 4737  df-res 4738  df-ima 4739  df-iota 5256  df-fun 5294  df-fn 5295  df-fv 5300
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