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Theorem fconstfv 5990
Description: A constant function expressed in terms of its functionality, domain, and value. See also fconst2 5984. (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 5626 . . 3  |-  ( F : A --> { B }  ->  F  Fn  A
)
2 fvconst 5957 . . . 4  |-  ( ( F : A --> { B }  /\  x  e.  A
)  ->  ( F `  x )  =  B )
32ralrimiva 2796 . . 3  |-  ( F : A --> { B }  ->  A. x  e.  A  ( F `  x )  =  B )
41, 3jca 520 . 2  |-  ( F : A --> { B }  ->  ( F  Fn  A  /\  A. x  e.  A  ( F `  x )  =  B ) )
5 fneq2 5570 . . . . . . 7  |-  ( A  =  (/)  ->  ( F  Fn  A  <->  F  Fn  (/) ) )
6 fn0 5599 . . . . . . 7  |-  ( F  Fn  (/)  <->  F  =  (/) )
75, 6syl6bb 254 . . . . . 6  |-  ( A  =  (/)  ->  ( F  Fn  A  <->  F  =  (/) ) )
8 f0 5662 . . . . . . 7  |-  (/) : (/) --> { B }
9 feq1 5611 . . . . . . 7  |-  ( F  =  (/)  ->  ( F : (/) --> { B }  <->  (/) :
(/) --> { B }
) )
108, 9mpbiri 226 . . . . . 6  |-  ( F  =  (/)  ->  F : (/) --> { B } )
117, 10syl6bi 221 . . . . 5  |-  ( A  =  (/)  ->  ( F  Fn  A  ->  F : (/) --> { B }
) )
12 feq2 5612 . . . . 5  |-  ( A  =  (/)  ->  ( F : A --> { B } 
<->  F : (/) --> { B } ) )
1311, 12sylibrd 227 . . . 4  |-  ( A  =  (/)  ->  ( F  Fn  A  ->  F : A --> { B }
) )
1413adantrd 456 . . 3  |-  ( A  =  (/)  ->  ( ( F  Fn  A  /\  A. x  e.  A  ( F `  x )  =  B )  ->  F : A --> { B } ) )
15 fvelrnb 5810 . . . . . . . . . 10  |-  ( F  Fn  A  ->  (
y  e.  ran  F  <->  E. z  e.  A  ( F `  z )  =  y ) )
16 fveq2 5763 . . . . . . . . . . . . . . 15  |-  ( x  =  z  ->  ( F `  x )  =  ( F `  z ) )
1716eqeq1d 2451 . . . . . . . . . . . . . 14  |-  ( x  =  z  ->  (
( F `  x
)  =  B  <->  ( F `  z )  =  B ) )
1817rspccva 3060 . . . . . . . . . . . . 13  |-  ( ( A. x  e.  A  ( F `  x )  =  B  /\  z  e.  A )  ->  ( F `  z )  =  B )
1918eqeq1d 2451 . . . . . . . . . . . 12  |-  ( ( A. x  e.  A  ( F `  x )  =  B  /\  z  e.  A )  ->  (
( F `  z
)  =  y  <->  B  =  y ) )
2019rexbidva 2729 . . . . . . . . . . 11  |-  ( A. x  e.  A  ( F `  x )  =  B  ->  ( E. z  e.  A  ( F `  z )  =  y  <->  E. z  e.  A  B  =  y ) )
21 r19.9rzv 3750 . . . . . . . . . . . 12  |-  ( A  =/=  (/)  ->  ( B  =  y  <->  E. z  e.  A  B  =  y )
)
2221bicomd 194 . . . . . . . . . . 11  |-  ( A  =/=  (/)  ->  ( E. z  e.  A  B  =  y  <->  B  =  y
) )
2320, 22sylan9bbr 683 . . . . . . . . . 10  |-  ( ( A  =/=  (/)  /\  A. x  e.  A  ( F `  x )  =  B )  ->  ( E. z  e.  A  ( F `  z )  =  y  <->  B  =  y ) )
2415, 23sylan9bbr 683 . . . . . . . . 9  |-  ( ( ( A  =/=  (/)  /\  A. x  e.  A  ( F `  x )  =  B )  /\  F  Fn  A )  ->  (
y  e.  ran  F  <->  B  =  y ) )
25 elsn 3858 . . . . . . . . . 10  |-  ( y  e.  { B }  <->  y  =  B )
26 eqcom 2445 . . . . . . . . . 10  |-  ( y  =  B  <->  B  =  y )
2725, 26bitr2i 243 . . . . . . . . 9  |-  ( B  =  y  <->  y  e.  { B } )
2824, 27syl6bb 254 . . . . . . . 8  |-  ( ( ( A  =/=  (/)  /\  A. x  e.  A  ( F `  x )  =  B )  /\  F  Fn  A )  ->  (
y  e.  ran  F  <->  y  e.  { B }
) )
2928eqrdv 2441 . . . . . . 7  |-  ( ( ( A  =/=  (/)  /\  A. x  e.  A  ( F `  x )  =  B )  /\  F  Fn  A )  ->  ran  F  =  { B }
)
3029an32s 781 . . . . . 6  |-  ( ( ( A  =/=  (/)  /\  F  Fn  A )  /\  A. x  e.  A  ( F `  x )  =  B )  ->  ran  F  =  { B }
)
3130exp31 589 . . . . 5  |-  ( A  =/=  (/)  ->  ( F  Fn  A  ->  ( A. x  e.  A  ( F `  x )  =  B  ->  ran  F  =  { B } ) ) )
3231imdistand 675 . . . 4  |-  ( A  =/=  (/)  ->  ( ( F  Fn  A  /\  A. x  e.  A  ( F `  x )  =  B )  -> 
( F  Fn  A  /\  ran  F  =  { B } ) ) )
33 df-fo 5495 . . . . 5  |-  ( F : A -onto-> { B } 
<->  ( F  Fn  A  /\  ran  F  =  { B } ) )
34 fof 5688 . . . . 5  |-  ( F : A -onto-> { B }  ->  F : A --> { B } )
3533, 34sylbir 206 . . . 4  |-  ( ( F  Fn  A  /\  ran  F  =  { B } )  ->  F : A --> { B }
)
3632, 35syl6 32 . . 3  |-  ( A  =/=  (/)  ->  ( ( F  Fn  A  /\  A. x  e.  A  ( F `  x )  =  B )  ->  F : A --> { B } ) )
3714, 36pm2.61ine 2687 . 2  |-  ( ( F  Fn  A  /\  A. x  e.  A  ( F `  x )  =  B )  ->  F : A --> { B } )
384, 37impbii 182 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 178    /\ wa 360    = wceq 1654    e. wcel 1728    =/= wne 2606   A.wral 2712   E.wrex 2713   (/)c0 3616   {csn 3843   ran crn 4914    Fn wfn 5484   -->wf 5485   -onto->wfo 5487   ` cfv 5489
This theorem is referenced by:  fconst3  5991  lnon0  22337  df0op2  23293  lfl1  30042
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1628  ax-9 1669  ax-8 1690  ax-14 1732  ax-6 1747  ax-7 1752  ax-11 1764  ax-12 1954  ax-ext 2424  ax-sep 4361  ax-nul 4369  ax-pr 4438
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1661  df-eu 2292  df-mo 2293  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-ral 2717  df-rex 2718  df-rab 2721  df-v 2967  df-sbc 3171  df-dif 3312  df-un 3314  df-in 3316  df-ss 3323  df-nul 3617  df-if 3768  df-sn 3849  df-pr 3850  df-op 3852  df-uni 4045  df-br 4244  df-opab 4298  df-mpt 4299  df-id 4533  df-xp 4919  df-rel 4920  df-cnv 4921  df-co 4922  df-dm 4923  df-rn 4924  df-iota 5453  df-fun 5491  df-fn 5492  df-f 5493  df-fo 5495  df-fv 5497
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