MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  fniniseg2 Structured version   Unicode version

Theorem fniniseg2 5845
Description: Inverse point images under functions expressed as abstractions. (Contributed by Stefan O'Rear, 1-Feb-2015.)
Assertion
Ref Expression
fniniseg2  |-  ( F  Fn  A  ->  ( `' F " { B } )  =  {
x  e.  A  | 
( F `  x
)  =  B }
)
Distinct variable groups:    x, A    x, F    x, B

Proof of Theorem fniniseg2
StepHypRef Expression
1 fncnvima2 5844 . 2  |-  ( F  Fn  A  ->  ( `' F " { B } )  =  {
x  e.  A  | 
( F `  x
)  e.  { B } } )
2 fvex 5734 . . . . 5  |-  ( F `
 x )  e. 
_V
32elsnc 3829 . . . 4  |-  ( ( F `  x )  e.  { B }  <->  ( F `  x )  =  B )
43a1i 11 . . 3  |-  ( x  e.  A  ->  (
( F `  x
)  e.  { B } 
<->  ( F `  x
)  =  B ) )
54rabbiia 2938 . 2  |-  { x  e.  A  |  ( F `  x )  e.  { B } }  =  { x  e.  A  |  ( F `  x )  =  B }
61, 5syl6eq 2483 1  |-  ( F  Fn  A  ->  ( `' F " { B } )  =  {
x  e.  A  | 
( F `  x
)  =  B }
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    = wceq 1652    e. wcel 1725   {crab 2701   {csn 3806   `'ccnv 4869   "cima 4873    Fn wfn 5441   ` cfv 5446
This theorem is referenced by:  idomrootle  27479  proot1hash  27487
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
  Copyright terms: Public domain W3C validator