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Theorem fconstfvm 5515
Description: A constant function expressed in terms of its functionality, domain, and value. See also fconst2 5514. (Contributed by Jim Kingdon, 8-Jan-2019.)
Assertion
Ref Expression
fconstfvm  |-  ( E. y  y  e.  A  ->  ( F : A --> { B }  <->  ( F  Fn  A  /\  A. x  e.  A  ( F `  x )  =  B ) ) )
Distinct variable groups:    x, A    x, B    x, F    y, A
Allowed substitution hints:    B( y)    F( y)

Proof of Theorem fconstfvm
Dummy variables  w  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ffn 5161 . . 3  |-  ( F : A --> { B }  ->  F  Fn  A
)
2 fvconst 5485 . . . 4  |-  ( ( F : A --> { B }  /\  x  e.  A
)  ->  ( F `  x )  =  B )
32ralrimiva 2446 . . 3  |-  ( F : A --> { B }  ->  A. x  e.  A  ( F `  x )  =  B )
41, 3jca 300 . 2  |-  ( F : A --> { B }  ->  ( F  Fn  A  /\  A. x  e.  A  ( F `  x )  =  B ) )
5 fvelrnb 5352 . . . . . . . . 9  |-  ( F  Fn  A  ->  (
w  e.  ran  F  <->  E. z  e.  A  ( F `  z )  =  w ) )
6 fveq2 5305 . . . . . . . . . . . . . 14  |-  ( x  =  z  ->  ( F `  x )  =  ( F `  z ) )
76eqeq1d 2096 . . . . . . . . . . . . 13  |-  ( x  =  z  ->  (
( F `  x
)  =  B  <->  ( F `  z )  =  B ) )
87rspccva 2721 . . . . . . . . . . . 12  |-  ( ( A. x  e.  A  ( F `  x )  =  B  /\  z  e.  A )  ->  ( F `  z )  =  B )
98eqeq1d 2096 . . . . . . . . . . 11  |-  ( ( A. x  e.  A  ( F `  x )  =  B  /\  z  e.  A )  ->  (
( F `  z
)  =  w  <->  B  =  w ) )
109rexbidva 2377 . . . . . . . . . 10  |-  ( A. x  e.  A  ( F `  x )  =  B  ->  ( E. z  e.  A  ( F `  z )  =  w  <->  E. z  e.  A  B  =  w ) )
11 r19.9rmv 3373 . . . . . . . . . . 11  |-  ( E. y  y  e.  A  ->  ( B  =  w  <->  E. z  e.  A  B  =  w )
)
1211bicomd 139 . . . . . . . . . 10  |-  ( E. y  y  e.  A  ->  ( E. z  e.  A  B  =  w  <-> 
B  =  w ) )
1310, 12sylan9bbr 451 . . . . . . . . 9  |-  ( ( E. y  y  e.  A  /\  A. x  e.  A  ( F `  x )  =  B )  ->  ( E. z  e.  A  ( F `  z )  =  w  <->  B  =  w
) )
145, 13sylan9bbr 451 . . . . . . . 8  |-  ( ( ( E. y  y  e.  A  /\  A. x  e.  A  ( F `  x )  =  B )  /\  F  Fn  A )  ->  (
w  e.  ran  F  <->  B  =  w ) )
15 velsn 3463 . . . . . . . . 9  |-  ( w  e.  { B }  <->  w  =  B )
16 eqcom 2090 . . . . . . . . 9  |-  ( w  =  B  <->  B  =  w )
1715, 16bitr2i 183 . . . . . . . 8  |-  ( B  =  w  <->  w  e.  { B } )
1814, 17syl6bb 194 . . . . . . 7  |-  ( ( ( E. y  y  e.  A  /\  A. x  e.  A  ( F `  x )  =  B )  /\  F  Fn  A )  ->  (
w  e.  ran  F  <->  w  e.  { B }
) )
1918eqrdv 2086 . . . . . 6  |-  ( ( ( E. y  y  e.  A  /\  A. x  e.  A  ( F `  x )  =  B )  /\  F  Fn  A )  ->  ran  F  =  { B }
)
2019an32s 535 . . . . 5  |-  ( ( ( E. y  y  e.  A  /\  F  Fn  A )  /\  A. x  e.  A  ( F `  x )  =  B )  ->  ran  F  =  { B }
)
2120exp31 356 . . . 4  |-  ( E. y  y  e.  A  ->  ( F  Fn  A  ->  ( A. x  e.  A  ( F `  x )  =  B  ->  ran  F  =  { B } ) ) )
2221imdistand 436 . . 3  |-  ( E. y  y  e.  A  ->  ( ( F  Fn  A  /\  A. x  e.  A  ( F `  x )  =  B )  ->  ( F  Fn  A  /\  ran  F  =  { B } ) ) )
23 df-fo 5021 . . . 4  |-  ( F : A -onto-> { B } 
<->  ( F  Fn  A  /\  ran  F  =  { B } ) )
24 fof 5233 . . . 4  |-  ( F : A -onto-> { B }  ->  F : A --> { B } )
2523, 24sylbir 133 . . 3  |-  ( ( F  Fn  A  /\  ran  F  =  { B } )  ->  F : A --> { B }
)
2622, 25syl6 33 . 2  |-  ( E. y  y  e.  A  ->  ( ( F  Fn  A  /\  A. x  e.  A  ( F `  x )  =  B )  ->  F : A
--> { B } ) )
274, 26impbid2 141 1  |-  ( E. y  y  e.  A  ->  ( F : A --> { B }  <->  ( F  Fn  A  /\  A. x  e.  A  ( F `  x )  =  B ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    <-> wb 103    = wceq 1289   E.wex 1426    e. wcel 1438   A.wral 2359   E.wrex 2360   {csn 3446   ran crn 4439    Fn wfn 5010   -->wf 5011   -onto->wfo 5013   ` cfv 5015
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3957  ax-pow 4009  ax-pr 4036
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-sbc 2841  df-un 3003  df-in 3005  df-ss 3012  df-pw 3431  df-sn 3452  df-pr 3453  df-op 3455  df-uni 3654  df-br 3846  df-opab 3900  df-mpt 3901  df-id 4120  df-xp 4444  df-rel 4445  df-cnv 4446  df-co 4447  df-dm 4448  df-rn 4449  df-iota 4980  df-fun 5017  df-fn 5018  df-f 5019  df-fo 5021  df-fv 5023
This theorem is referenced by:  fconst3m  5516
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