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Theorem fncnvima2 5996
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 5994 . . 3  |-  ( F  Fn  A  ->  (
x  e.  ( `' F " B )  <-> 
( x  e.  A  /\  ( F `  x
)  e.  B ) ) )
21abbi2dv 2599 . 2  |-  ( F  Fn  A  ->  ( `' F " B )  =  { x  |  ( x  e.  A  /\  ( F `  x
)  e.  B ) } )
3 df-rab 2818 . 2  |-  { x  e.  A  |  ( F `  x )  e.  B }  =  {
x  |  ( x  e.  A  /\  ( F `  x )  e.  B ) }
42, 3syl6eqr 2521 1  |-  ( F  Fn  A  ->  ( `' F " B )  =  { x  e.  A  |  ( F `
 x )  e.  B } )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1374    e. wcel 1762   {cab 2447   {crab 2813   `'ccnv 4993   "cima 4997    Fn wfn 5576   ` cfv 5581
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1963  ax-ext 2440  ax-sep 4563  ax-nul 4571  ax-pr 4681
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2274  df-mo 2275  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2612  df-ne 2659  df-ral 2814  df-rex 2815  df-rab 2818  df-v 3110  df-sbc 3327  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3781  df-if 3935  df-sn 4023  df-pr 4025  df-op 4029  df-uni 4241  df-br 4443  df-opab 4501  df-id 4790  df-xp 5000  df-rel 5001  df-cnv 5002  df-co 5003  df-dm 5004  df-rn 5005  df-res 5006  df-ima 5007  df-iota 5544  df-fun 5583  df-fn 5584  df-fv 5589
This theorem is referenced by:  fniniseg2  5997  fnniniseg2OLD  5998  r0cld  19969  iunpreima  27093  xppreima  27147  xpinpreima  27512  xpinpreima2  27513  orvcval2  28025
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