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Theorem fncnvima2 5987
Description: Inverse images under functions expressed as abstractions. (Contributed by Stefan O'Rear, 1-Feb-2015.)
Assertion
Ref Expression
fncnvima2  |-  ( F  Fn  A  ->  ( `' F " B )  =  { x  e.  A  |  ( F `
 x )  e.  B } )
Distinct variable groups:    x, A    x, F    x, B

Proof of Theorem fncnvima2
StepHypRef Expression
1 elpreima 5985 . . 3  |-  ( F  Fn  A  ->  (
x  e.  ( `' F " B )  <-> 
( x  e.  A  /\  ( F `  x
)  e.  B ) ) )
21abbi2dv 2539 . 2  |-  ( F  Fn  A  ->  ( `' F " B )  =  { x  |  ( x  e.  A  /\  ( F `  x
)  e.  B ) } )
3 df-rab 2763 . 2  |-  { x  e.  A  |  ( F `  x )  e.  B }  =  {
x  |  ( x  e.  A  /\  ( F `  x )  e.  B ) }
42, 3syl6eqr 2461 1  |-  ( F  Fn  A  ->  ( `' F " B )  =  { x  e.  A  |  ( F `
 x )  e.  B } )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    = wceq 1405    e. wcel 1842   {cab 2387   {crab 2758   `'ccnv 4822   "cima 4826    Fn wfn 5564   ` cfv 5569
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 4517  ax-nul 4525  ax-pr 4630
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 2759  df-rex 2760  df-rab 2763  df-v 3061  df-sbc 3278  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-nul 3739  df-if 3886  df-sn 3973  df-pr 3975  df-op 3979  df-uni 4192  df-br 4396  df-opab 4454  df-id 4738  df-xp 4829  df-rel 4830  df-cnv 4831  df-co 4832  df-dm 4833  df-rn 4834  df-res 4835  df-ima 4836  df-iota 5533  df-fun 5571  df-fn 5572  df-fv 5577
This theorem is referenced by:  fniniseg2  5988  fnniniseg2OLD  5989  fncnvimaeqv  15713  r0cld  20531  iunpreima  27862  xppreima  27930  xpinpreima  28341  xpinpreima2  28342  orvcval2  28903
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