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

Theorem fconstfv 6132
Description: A constant function expressed in terms of its functionality, domain, and value. See also fconst2 6127. (Contributed by NM, 27-Aug-2004.) (Proof shortened by OpenAI, 25-Mar-2020.)
Assertion
Ref Expression
fconstfv  |-  ( F : A --> { B } 
<->  ( F  Fn  A  /\  A. x  e.  A  ( F `  x )  =  B ) )
Distinct variable groups:    x, A    x, B    x, F

Proof of Theorem fconstfv
StepHypRef Expression
1 ffnfv 6055 . 2  |-  ( F : A --> { B } 
<->  ( F  Fn  A  /\  A. x  e.  A  ( F `  x )  e.  { B }
) )
2 fvex 5882 . . . . 5  |-  ( F `
 x )  e. 
_V
32elsnc 4017 . . . 4  |-  ( ( F `  x )  e.  { B }  <->  ( F `  x )  =  B )
43ralbii 2854 . . 3  |-  ( A. x  e.  A  ( F `  x )  e.  { B }  <->  A. x  e.  A  ( F `  x )  =  B )
54anbi2i 698 . 2  |-  ( ( F  Fn  A  /\  A. x  e.  A  ( F `  x )  e.  { B }
)  <->  ( F  Fn  A  /\  A. x  e.  A  ( F `  x )  =  B ) )
61, 5bitri 252 1  |-  ( F : A --> { B } 
<->  ( F  Fn  A  /\  A. x  e.  A  ( F `  x )  =  B ) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 187    /\ wa 370    = wceq 1437    e. wcel 1867   A.wral 2773   {csn 3993    Fn wfn 5587   -->wf 5588   ` cfv 5592
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1838  ax-9 1871  ax-10 1886  ax-11 1891  ax-12 1904  ax-13 2052  ax-ext 2398  ax-sep 4539  ax-nul 4547  ax-pr 4652
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2267  df-mo 2268  df-clab 2406  df-cleq 2412  df-clel 2415  df-nfc 2570  df-ne 2618  df-ral 2778  df-rex 2779  df-rab 2782  df-v 3080  df-sbc 3297  df-dif 3436  df-un 3438  df-in 3440  df-ss 3447  df-nul 3759  df-if 3907  df-sn 3994  df-pr 3996  df-op 4000  df-uni 4214  df-br 4418  df-opab 4476  df-mpt 4477  df-id 4760  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-iota 5556  df-fun 5594  df-fn 5595  df-f 5596  df-fv 5600
This theorem is referenced by:  fconst3  6134  repsdf2  12855  rrxcph  22270  lnon0  26325  df0op2  27281  poimir  31721  lfl1  32389
  Copyright terms: Public domain W3C validator