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Theorem fconstfv 5750
Description: A constant function expressed in terms of its functionality, domain, and value. See also fconst2 5746. (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 5405 . . 3  |-  ( F : A --> { B }  ->  F  Fn  A
)
2 fvconst 5724 . . . 4  |-  ( ( F : A --> { B }  /\  x  e.  A
)  ->  ( F `  x )  =  B )
32ralrimiva 2639 . . 3  |-  ( F : A --> { B }  ->  A. x  e.  A  ( F `  x )  =  B )
41, 3jca 518 . 2  |-  ( F : A --> { B }  ->  ( F  Fn  A  /\  A. x  e.  A  ( F `  x )  =  B ) )
5 fneq2 5350 . . . . . . 7  |-  ( A  =  (/)  ->  ( F  Fn  A  <->  F  Fn  (/) ) )
6 fn0 5379 . . . . . . 7  |-  ( F  Fn  (/)  <->  F  =  (/) )
75, 6syl6bb 252 . . . . . 6  |-  ( A  =  (/)  ->  ( F  Fn  A  <->  F  =  (/) ) )
8 f0 5441 . . . . . . 7  |-  (/) : (/) --> { B }
9 feq1 5391 . . . . . . 7  |-  ( F  =  (/)  ->  ( F : (/) --> { B }  <->  (/) :
(/) --> { B }
) )
108, 9mpbiri 224 . . . . . 6  |-  ( F  =  (/)  ->  F : (/) --> { B } )
117, 10syl6bi 219 . . . . 5  |-  ( A  =  (/)  ->  ( F  Fn  A  ->  F : (/) --> { B }
) )
12 feq2 5392 . . . . 5  |-  ( A  =  (/)  ->  ( F : A --> { B } 
<->  F : (/) --> { B } ) )
1311, 12sylibrd 225 . . . 4  |-  ( A  =  (/)  ->  ( F  Fn  A  ->  F : A --> { B }
) )
1413adantrd 454 . . 3  |-  ( A  =  (/)  ->  ( ( F  Fn  A  /\  A. x  e.  A  ( F `  x )  =  B )  ->  F : A --> { B } ) )
15 fvelrnb 5586 . . . . . . . . . 10  |-  ( F  Fn  A  ->  (
y  e.  ran  F  <->  E. z  e.  A  ( F `  z )  =  y ) )
16 fveq2 5541 . . . . . . . . . . . . . . 15  |-  ( x  =  z  ->  ( F `  x )  =  ( F `  z ) )
1716eqeq1d 2304 . . . . . . . . . . . . . 14  |-  ( x  =  z  ->  (
( F `  x
)  =  B  <->  ( F `  z )  =  B ) )
1817rspccva 2896 . . . . . . . . . . . . 13  |-  ( ( A. x  e.  A  ( F `  x )  =  B  /\  z  e.  A )  ->  ( F `  z )  =  B )
1918eqeq1d 2304 . . . . . . . . . . . 12  |-  ( ( A. x  e.  A  ( F `  x )  =  B  /\  z  e.  A )  ->  (
( F `  z
)  =  y  <->  B  =  y ) )
2019rexbidva 2573 . . . . . . . . . . 11  |-  ( A. x  e.  A  ( F `  x )  =  B  ->  ( E. z  e.  A  ( F `  z )  =  y  <->  E. z  e.  A  B  =  y ) )
21 r19.9rzv 3561 . . . . . . . . . . . 12  |-  ( A  =/=  (/)  ->  ( B  =  y  <->  E. z  e.  A  B  =  y )
)
2221bicomd 192 . . . . . . . . . . 11  |-  ( A  =/=  (/)  ->  ( E. z  e.  A  B  =  y  <->  B  =  y
) )
2320, 22sylan9bbr 681 . . . . . . . . . 10  |-  ( ( A  =/=  (/)  /\  A. x  e.  A  ( F `  x )  =  B )  ->  ( E. z  e.  A  ( F `  z )  =  y  <->  B  =  y ) )
2415, 23sylan9bbr 681 . . . . . . . . 9  |-  ( ( ( A  =/=  (/)  /\  A. x  e.  A  ( F `  x )  =  B )  /\  F  Fn  A )  ->  (
y  e.  ran  F  <->  B  =  y ) )
25 elsn 3668 . . . . . . . . . 10  |-  ( y  e.  { B }  <->  y  =  B )
26 eqcom 2298 . . . . . . . . . 10  |-  ( y  =  B  <->  B  =  y )
2725, 26bitr2i 241 . . . . . . . . 9  |-  ( B  =  y  <->  y  e.  { B } )
2824, 27syl6bb 252 . . . . . . . 8  |-  ( ( ( A  =/=  (/)  /\  A. x  e.  A  ( F `  x )  =  B )  /\  F  Fn  A )  ->  (
y  e.  ran  F  <->  y  e.  { B }
) )
2928eqrdv 2294 . . . . . . 7  |-  ( ( ( A  =/=  (/)  /\  A. x  e.  A  ( F `  x )  =  B )  /\  F  Fn  A )  ->  ran  F  =  { B }
)
3029an32s 779 . . . . . 6  |-  ( ( ( A  =/=  (/)  /\  F  Fn  A )  /\  A. x  e.  A  ( F `  x )  =  B )  ->  ran  F  =  { B }
)
3130exp31 587 . . . . 5  |-  ( A  =/=  (/)  ->  ( F  Fn  A  ->  ( A. x  e.  A  ( F `  x )  =  B  ->  ran  F  =  { B } ) ) )
3231imdistand 673 . . . 4  |-  ( A  =/=  (/)  ->  ( ( F  Fn  A  /\  A. x  e.  A  ( F `  x )  =  B )  -> 
( F  Fn  A  /\  ran  F  =  { B } ) ) )
33 df-fo 5277 . . . . 5  |-  ( F : A -onto-> { B } 
<->  ( F  Fn  A  /\  ran  F  =  { B } ) )
34 fof 5467 . . . . 5  |-  ( F : A -onto-> { B }  ->  F : A --> { B } )
3533, 34sylbir 204 . . . 4  |-  ( ( F  Fn  A  /\  ran  F  =  { B } )  ->  F : A --> { B }
)
3632, 35syl6 29 . . 3  |-  ( A  =/=  (/)  ->  ( ( F  Fn  A  /\  A. x  e.  A  ( F `  x )  =  B )  ->  F : A --> { B } ) )
3714, 36pm2.61ine 2535 . 2  |-  ( ( F  Fn  A  /\  A. x  e.  A  ( F `  x )  =  B )  ->  F : A --> { B } )
384, 37impbii 180 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 176    /\ wa 358    = wceq 1632    e. wcel 1696    =/= wne 2459   A.wral 2556   E.wrex 2557   (/)c0 3468   {csn 3653   ran crn 4706    Fn wfn 5266   -->wf 5267   -onto->wfo 5269   ` cfv 5271
This theorem is referenced by:  fconst3  5751  lnon0  21392  df0op2  22348  lfl1  29882
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-fo 5277  df-fv 5279
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