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

Theorem fconstfv 5954
Description: A constant function expressed in terms of its functionality, domain, and value. See also fconst2 5948. (Contributed by NM, 27-Aug-2004.)
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
Dummy variables  y 
z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ffn 5591 . . 3  |-  ( F : A --> { B }  ->  F  Fn  A
)
2 fvconst 5921 . . . 4  |-  ( ( F : A --> { B }  /\  x  e.  A
)  ->  ( F `  x )  =  B )
32ralrimiva 2789 . . 3  |-  ( F : A --> { B }  ->  A. x  e.  A  ( F `  x )  =  B )
41, 3jca 519 . 2  |-  ( F : A --> { B }  ->  ( F  Fn  A  /\  A. x  e.  A  ( F `  x )  =  B ) )
5 fneq2 5535 . . . . . . 7  |-  ( A  =  (/)  ->  ( F  Fn  A  <->  F  Fn  (/) ) )
6 fn0 5564 . . . . . . 7  |-  ( F  Fn  (/)  <->  F  =  (/) )
75, 6syl6bb 253 . . . . . 6  |-  ( A  =  (/)  ->  ( F  Fn  A  <->  F  =  (/) ) )
8 f0 5627 . . . . . . 7  |-  (/) : (/) --> { B }
9 feq1 5576 . . . . . . 7  |-  ( F  =  (/)  ->  ( F : (/) --> { B }  <->  (/) :
(/) --> { B }
) )
108, 9mpbiri 225 . . . . . 6  |-  ( F  =  (/)  ->  F : (/) --> { B } )
117, 10syl6bi 220 . . . . 5  |-  ( A  =  (/)  ->  ( F  Fn  A  ->  F : (/) --> { B }
) )
12 feq2 5577 . . . . 5  |-  ( A  =  (/)  ->  ( F : A --> { B } 
<->  F : (/) --> { B } ) )
1311, 12sylibrd 226 . . . 4  |-  ( A  =  (/)  ->  ( F  Fn  A  ->  F : A --> { B }
) )
1413adantrd 455 . . 3  |-  ( A  =  (/)  ->  ( ( F  Fn  A  /\  A. x  e.  A  ( F `  x )  =  B )  ->  F : A --> { B } ) )
15 fvelrnb 5774 . . . . . . . . . 10  |-  ( F  Fn  A  ->  (
y  e.  ran  F  <->  E. z  e.  A  ( F `  z )  =  y ) )
16 fveq2 5728 . . . . . . . . . . . . . . 15  |-  ( x  =  z  ->  ( F `  x )  =  ( F `  z ) )
1716eqeq1d 2444 . . . . . . . . . . . . . 14  |-  ( x  =  z  ->  (
( F `  x
)  =  B  <->  ( F `  z )  =  B ) )
1817rspccva 3051 . . . . . . . . . . . . 13  |-  ( ( A. x  e.  A  ( F `  x )  =  B  /\  z  e.  A )  ->  ( F `  z )  =  B )
1918eqeq1d 2444 . . . . . . . . . . . 12  |-  ( ( A. x  e.  A  ( F `  x )  =  B  /\  z  e.  A )  ->  (
( F `  z
)  =  y  <->  B  =  y ) )
2019rexbidva 2722 . . . . . . . . . . 11  |-  ( A. x  e.  A  ( F `  x )  =  B  ->  ( E. z  e.  A  ( F `  z )  =  y  <->  E. z  e.  A  B  =  y ) )
21 r19.9rzv 3722 . . . . . . . . . . . 12  |-  ( A  =/=  (/)  ->  ( B  =  y  <->  E. z  e.  A  B  =  y )
)
2221bicomd 193 . . . . . . . . . . 11  |-  ( A  =/=  (/)  ->  ( E. z  e.  A  B  =  y  <->  B  =  y
) )
2320, 22sylan9bbr 682 . . . . . . . . . 10  |-  ( ( A  =/=  (/)  /\  A. x  e.  A  ( F `  x )  =  B )  ->  ( E. z  e.  A  ( F `  z )  =  y  <->  B  =  y ) )
2415, 23sylan9bbr 682 . . . . . . . . 9  |-  ( ( ( A  =/=  (/)  /\  A. x  e.  A  ( F `  x )  =  B )  /\  F  Fn  A )  ->  (
y  e.  ran  F  <->  B  =  y ) )
25 elsn 3829 . . . . . . . . . 10  |-  ( y  e.  { B }  <->  y  =  B )
26 eqcom 2438 . . . . . . . . . 10  |-  ( y  =  B  <->  B  =  y )
2725, 26bitr2i 242 . . . . . . . . 9  |-  ( B  =  y  <->  y  e.  { B } )
2824, 27syl6bb 253 . . . . . . . 8  |-  ( ( ( A  =/=  (/)  /\  A. x  e.  A  ( F `  x )  =  B )  /\  F  Fn  A )  ->  (
y  e.  ran  F  <->  y  e.  { B }
) )
2928eqrdv 2434 . . . . . . 7  |-  ( ( ( A  =/=  (/)  /\  A. x  e.  A  ( F `  x )  =  B )  /\  F  Fn  A )  ->  ran  F  =  { B }
)
3029an32s 780 . . . . . 6  |-  ( ( ( A  =/=  (/)  /\  F  Fn  A )  /\  A. x  e.  A  ( F `  x )  =  B )  ->  ran  F  =  { B }
)
3130exp31 588 . . . . 5  |-  ( A  =/=  (/)  ->  ( F  Fn  A  ->  ( A. x  e.  A  ( F `  x )  =  B  ->  ran  F  =  { B } ) ) )
3231imdistand 674 . . . 4  |-  ( A  =/=  (/)  ->  ( ( F  Fn  A  /\  A. x  e.  A  ( F `  x )  =  B )  -> 
( F  Fn  A  /\  ran  F  =  { B } ) ) )
33 df-fo 5460 . . . . 5  |-  ( F : A -onto-> { B } 
<->  ( F  Fn  A  /\  ran  F  =  { B } ) )
34 fof 5653 . . . . 5  |-  ( F : A -onto-> { B }  ->  F : A --> { B } )
3533, 34sylbir 205 . . . 4  |-  ( ( F  Fn  A  /\  ran  F  =  { B } )  ->  F : A --> { B }
)
3632, 35syl6 31 . . 3  |-  ( A  =/=  (/)  ->  ( ( F  Fn  A  /\  A. x  e.  A  ( F `  x )  =  B )  ->  F : A --> { B } ) )
3714, 36pm2.61ine 2680 . 2  |-  ( ( F  Fn  A  /\  A. x  e.  A  ( F `  x )  =  B )  ->  F : A --> { B } )
384, 37impbii 181 1  |-  ( F : A --> { B } 
<->  ( F  Fn  A  /\  A. x  e.  A  ( F `  x )  =  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1652    e. wcel 1725    =/= wne 2599   A.wral 2705   E.wrex 2706   (/)c0 3628   {csn 3814   ran crn 4879    Fn wfn 5449   -->wf 5450   -onto->wfo 5452   ` cfv 5454
This theorem is referenced by:  fconst3  5955  lnon0  22299  df0op2  23255  lfl1  29868
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 2417  ax-sep 4330  ax-nul 4338  ax-pr 4403
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 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-sbc 3162  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-fo 5460  df-fv 5462
  Copyright terms: Public domain W3C validator