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Theorem fncnvima2 5844
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 5842 . . 3  |-  ( F  Fn  A  ->  (
x  e.  ( `' F " B )  <-> 
( x  e.  A  /\  ( F `  x
)  e.  B ) ) )
21abbi2dv 2550 . 2  |-  ( F  Fn  A  ->  ( `' F " B )  =  { x  |  ( x  e.  A  /\  ( F `  x
)  e.  B ) } )
3 df-rab 2706 . 2  |-  { x  e.  A  |  ( F `  x )  e.  B }  =  {
x  |  ( x  e.  A  /\  ( F `  x )  e.  B ) }
42, 3syl6eqr 2485 1  |-  ( F  Fn  A  ->  ( `' F " B )  =  { x  e.  A  |  ( F `
 x )  e.  B } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725   {cab 2421   {crab 2701   `'ccnv 4869   "cima 4873    Fn wfn 5441   ` cfv 5446
This theorem is referenced by:  fniniseg2  5845  fnniniseg2  5846  r0cld  17762  xppreima  24051  xpinpreima  24296  xpinpreima2  24297  orvcval2  24708
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pr 4395
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-fv 5454
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