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Theorem fconstfvOLD 6109
Description: A constant function expressed in terms of its functionality, domain, and value. See also fconst2 6104. (Contributed by NM, 27-Aug-2004.) Obsolete version of fconstfv 6108 as of 25-Mar-2020. (New usage is discouraged.) (Proof modification is discouraged.)
Assertion
Ref Expression
fconstfvOLD  |-  ( 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 fconstfvOLD
Dummy variables  y 
z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ffn 5713 . . 3  |-  ( F : A --> { B }  ->  F  Fn  A
)
2 fvconst 6065 . . . 4  |-  ( ( F : A --> { B }  /\  x  e.  A
)  ->  ( F `  x )  =  B )
32ralrimiva 2868 . . 3  |-  ( F : A --> { B }  ->  A. x  e.  A  ( F `  x )  =  B )
41, 3jca 530 . 2  |-  ( F : A --> { B }  ->  ( F  Fn  A  /\  A. x  e.  A  ( F `  x )  =  B ) )
5 fneq2 5652 . . . . . . 7  |-  ( A  =  (/)  ->  ( F  Fn  A  <->  F  Fn  (/) ) )
6 fn0 5682 . . . . . . 7  |-  ( F  Fn  (/)  <->  F  =  (/) )
75, 6syl6bb 261 . . . . . 6  |-  ( A  =  (/)  ->  ( F  Fn  A  <->  F  =  (/) ) )
8 f0 5748 . . . . . . 7  |-  (/) : (/) --> { B }
9 feq1 5695 . . . . . . 7  |-  ( F  =  (/)  ->  ( F : (/) --> { B }  <->  (/) :
(/) --> { B }
) )
108, 9mpbiri 233 . . . . . 6  |-  ( F  =  (/)  ->  F : (/) --> { B } )
117, 10syl6bi 228 . . . . 5  |-  ( A  =  (/)  ->  ( F  Fn  A  ->  F : (/) --> { B }
) )
12 feq2 5696 . . . . 5  |-  ( A  =  (/)  ->  ( F : A --> { B } 
<->  F : (/) --> { B } ) )
1311, 12sylibrd 234 . . . 4  |-  ( A  =  (/)  ->  ( F  Fn  A  ->  F : A --> { B }
) )
1413adantrd 466 . . 3  |-  ( A  =  (/)  ->  ( ( F  Fn  A  /\  A. x  e.  A  ( F `  x )  =  B )  ->  F : A --> { B } ) )
15 fvelrnb 5895 . . . . . . . . . 10  |-  ( F  Fn  A  ->  (
y  e.  ran  F  <->  E. z  e.  A  ( F `  z )  =  y ) )
16 fveq2 5848 . . . . . . . . . . . . . . 15  |-  ( x  =  z  ->  ( F `  x )  =  ( F `  z ) )
1716eqeq1d 2456 . . . . . . . . . . . . . 14  |-  ( x  =  z  ->  (
( F `  x
)  =  B  <->  ( F `  z )  =  B ) )
1817rspccva 3206 . . . . . . . . . . . . 13  |-  ( ( A. x  e.  A  ( F `  x )  =  B  /\  z  e.  A )  ->  ( F `  z )  =  B )
1918eqeq1d 2456 . . . . . . . . . . . 12  |-  ( ( A. x  e.  A  ( F `  x )  =  B  /\  z  e.  A )  ->  (
( F `  z
)  =  y  <->  B  =  y ) )
2019rexbidva 2962 . . . . . . . . . . 11  |-  ( A. x  e.  A  ( F `  x )  =  B  ->  ( E. z  e.  A  ( F `  z )  =  y  <->  E. z  e.  A  B  =  y ) )
21 r19.9rzv 3911 . . . . . . . . . . . 12  |-  ( A  =/=  (/)  ->  ( B  =  y  <->  E. z  e.  A  B  =  y )
)
2221bicomd 201 . . . . . . . . . . 11  |-  ( A  =/=  (/)  ->  ( E. z  e.  A  B  =  y  <->  B  =  y
) )
2320, 22sylan9bbr 698 . . . . . . . . . 10  |-  ( ( A  =/=  (/)  /\  A. x  e.  A  ( F `  x )  =  B )  ->  ( E. z  e.  A  ( F `  z )  =  y  <->  B  =  y ) )
2415, 23sylan9bbr 698 . . . . . . . . 9  |-  ( ( ( A  =/=  (/)  /\  A. x  e.  A  ( F `  x )  =  B )  /\  F  Fn  A )  ->  (
y  e.  ran  F  <->  B  =  y ) )
25 elsn 4030 . . . . . . . . . 10  |-  ( y  e.  { B }  <->  y  =  B )
26 eqcom 2463 . . . . . . . . . 10  |-  ( y  =  B  <->  B  =  y )
2725, 26bitr2i 250 . . . . . . . . 9  |-  ( B  =  y  <->  y  e.  { B } )
2824, 27syl6bb 261 . . . . . . . 8  |-  ( ( ( A  =/=  (/)  /\  A. x  e.  A  ( F `  x )  =  B )  /\  F  Fn  A )  ->  (
y  e.  ran  F  <->  y  e.  { B }
) )
2928eqrdv 2451 . . . . . . 7  |-  ( ( ( A  =/=  (/)  /\  A. x  e.  A  ( F `  x )  =  B )  /\  F  Fn  A )  ->  ran  F  =  { B }
)
3029an32s 802 . . . . . 6  |-  ( ( ( A  =/=  (/)  /\  F  Fn  A )  /\  A. x  e.  A  ( F `  x )  =  B )  ->  ran  F  =  { B }
)
3130exp31 602 . . . . 5  |-  ( A  =/=  (/)  ->  ( F  Fn  A  ->  ( A. x  e.  A  ( F `  x )  =  B  ->  ran  F  =  { B } ) ) )
3231imdistand 690 . . . 4  |-  ( A  =/=  (/)  ->  ( ( F  Fn  A  /\  A. x  e.  A  ( F `  x )  =  B )  -> 
( F  Fn  A  /\  ran  F  =  { B } ) ) )
33 df-fo 5576 . . . . 5  |-  ( F : A -onto-> { B } 
<->  ( F  Fn  A  /\  ran  F  =  { B } ) )
34 fof 5777 . . . . 5  |-  ( F : A -onto-> { B }  ->  F : A --> { B } )
3533, 34sylbir 213 . . . 4  |-  ( ( F  Fn  A  /\  ran  F  =  { B } )  ->  F : A --> { B }
)
3632, 35syl6 33 . . 3  |-  ( A  =/=  (/)  ->  ( ( F  Fn  A  /\  A. x  e.  A  ( F `  x )  =  B )  ->  F : A --> { B } ) )
3714, 36pm2.61ine 2767 . 2  |-  ( ( F  Fn  A  /\  A. x  e.  A  ( F `  x )  =  B )  ->  F : A --> { B } )
384, 37impbii 188 1  |-  ( F : A --> { B } 
<->  ( F  Fn  A  /\  A. x  e.  A  ( F `  x )  =  B ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367    = wceq 1398    e. wcel 1823    =/= wne 2649   A.wral 2804   E.wrex 2805   (/)c0 3783   {csn 4016   ran crn 4989    Fn wfn 5565   -->wf 5566   -onto->wfo 5568   ` cfv 5570
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pr 4676
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-sbc 3325  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-br 4440  df-opab 4498  df-mpt 4499  df-id 4784  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-fo 5576  df-fv 5578
This theorem is referenced by: (None)
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