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Theorem fconstfvm 5755
Description: A constant function expressed in terms of its functionality, domain, and value. See also fconst2 5754. (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 5384 . . 3  |-  ( F : A --> { B }  ->  F  Fn  A
)
2 fvconst 5725 . . . 4  |-  ( ( F : A --> { B }  /\  x  e.  A
)  ->  ( F `  x )  =  B )
32ralrimiva 2563 . . 3  |-  ( F : A --> { B }  ->  A. x  e.  A  ( F `  x )  =  B )
41, 3jca 306 . 2  |-  ( F : A --> { B }  ->  ( F  Fn  A  /\  A. x  e.  A  ( F `  x )  =  B ) )
5 fvelrnb 5584 . . . . . . . . 9  |-  ( F  Fn  A  ->  (
w  e.  ran  F  <->  E. z  e.  A  ( F `  z )  =  w ) )
6 fveq2 5534 . . . . . . . . . . . . . 14  |-  ( x  =  z  ->  ( F `  x )  =  ( F `  z ) )
76eqeq1d 2198 . . . . . . . . . . . . 13  |-  ( x  =  z  ->  (
( F `  x
)  =  B  <->  ( F `  z )  =  B ) )
87rspccva 2855 . . . . . . . . . . . 12  |-  ( ( A. x  e.  A  ( F `  x )  =  B  /\  z  e.  A )  ->  ( F `  z )  =  B )
98eqeq1d 2198 . . . . . . . . . . 11  |-  ( ( A. x  e.  A  ( F `  x )  =  B  /\  z  e.  A )  ->  (
( F `  z
)  =  w  <->  B  =  w ) )
109rexbidva 2487 . . . . . . . . . 10  |-  ( A. x  e.  A  ( F `  x )  =  B  ->  ( E. z  e.  A  ( F `  z )  =  w  <->  E. z  e.  A  B  =  w ) )
11 r19.9rmv 3529 . . . . . . . . . . 11  |-  ( E. y  y  e.  A  ->  ( B  =  w  <->  E. z  e.  A  B  =  w )
)
1211bicomd 141 . . . . . . . . . 10  |-  ( E. y  y  e.  A  ->  ( E. z  e.  A  B  =  w  <-> 
B  =  w ) )
1310, 12sylan9bbr 463 . . . . . . . . 9  |-  ( ( E. y  y  e.  A  /\  A. x  e.  A  ( F `  x )  =  B )  ->  ( E. z  e.  A  ( F `  z )  =  w  <->  B  =  w
) )
145, 13sylan9bbr 463 . . . . . . . 8  |-  ( ( ( E. y  y  e.  A  /\  A. x  e.  A  ( F `  x )  =  B )  /\  F  Fn  A )  ->  (
w  e.  ran  F  <->  B  =  w ) )
15 velsn 3624 . . . . . . . . 9  |-  ( w  e.  { B }  <->  w  =  B )
16 eqcom 2191 . . . . . . . . 9  |-  ( w  =  B  <->  B  =  w )
1715, 16bitr2i 185 . . . . . . . 8  |-  ( B  =  w  <->  w  e.  { B } )
1814, 17bitrdi 196 . . . . . . 7  |-  ( ( ( E. y  y  e.  A  /\  A. x  e.  A  ( F `  x )  =  B )  /\  F  Fn  A )  ->  (
w  e.  ran  F  <->  w  e.  { B }
) )
1918eqrdv 2187 . . . . . 6  |-  ( ( ( E. y  y  e.  A  /\  A. x  e.  A  ( F `  x )  =  B )  /\  F  Fn  A )  ->  ran  F  =  { B }
)
2019an32s 568 . . . . 5  |-  ( ( ( E. y  y  e.  A  /\  F  Fn  A )  /\  A. x  e.  A  ( F `  x )  =  B )  ->  ran  F  =  { B }
)
2120exp31 364 . . . 4  |-  ( E. y  y  e.  A  ->  ( F  Fn  A  ->  ( A. x  e.  A  ( F `  x )  =  B  ->  ran  F  =  { B } ) ) )
2221imdistand 447 . . 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 5241 . . . 4  |-  ( F : A -onto-> { B } 
<->  ( F  Fn  A  /\  ran  F  =  { B } ) )
24 fof 5457 . . . 4  |-  ( F : A -onto-> { B }  ->  F : A --> { B } )
2523, 24sylbir 135 . . 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 143 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 104    <-> wb 105    = wceq 1364   E.wex 1503    e. wcel 2160   A.wral 2468   E.wrex 2469   {csn 3607   ran crn 4645    Fn wfn 5230   -->wf 5231   -onto->wfo 5233   ` cfv 5235
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4192  ax-pr 4227
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-v 2754  df-sbc 2978  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-br 4019  df-opab 4080  df-mpt 4081  df-id 4311  df-xp 4650  df-rel 4651  df-cnv 4652  df-co 4653  df-dm 4654  df-rn 4655  df-iota 5196  df-fun 5237  df-fn 5238  df-f 5239  df-fo 5241  df-fv 5243
This theorem is referenced by:  fconst3m  5756
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