ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  fconstfvm Unicode version

Theorem fconstfvm 5407
Description: A constant function expressed in terms of its functionality, domain, and value. See also fconst2 5406. (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 5074 . . 3  |-  ( F : A --> { B }  ->  F  Fn  A
)
2 fvconst 5379 . . . 4  |-  ( ( F : A --> { B }  /\  x  e.  A
)  ->  ( F `  x )  =  B )
32ralrimiva 2409 . . 3  |-  ( F : A --> { B }  ->  A. x  e.  A  ( F `  x )  =  B )
41, 3jca 294 . 2  |-  ( F : A --> { B }  ->  ( F  Fn  A  /\  A. x  e.  A  ( F `  x )  =  B ) )
5 fvelrnb 5249 . . . . . . . . 9  |-  ( F  Fn  A  ->  (
w  e.  ran  F  <->  E. z  e.  A  ( F `  z )  =  w ) )
6 fveq2 5206 . . . . . . . . . . . . . 14  |-  ( x  =  z  ->  ( F `  x )  =  ( F `  z ) )
76eqeq1d 2064 . . . . . . . . . . . . 13  |-  ( x  =  z  ->  (
( F `  x
)  =  B  <->  ( F `  z )  =  B ) )
87rspccva 2672 . . . . . . . . . . . 12  |-  ( ( A. x  e.  A  ( F `  x )  =  B  /\  z  e.  A )  ->  ( F `  z )  =  B )
98eqeq1d 2064 . . . . . . . . . . 11  |-  ( ( A. x  e.  A  ( F `  x )  =  B  /\  z  e.  A )  ->  (
( F `  z
)  =  w  <->  B  =  w ) )
109rexbidva 2340 . . . . . . . . . 10  |-  ( A. x  e.  A  ( F `  x )  =  B  ->  ( E. z  e.  A  ( F `  z )  =  w  <->  E. z  e.  A  B  =  w ) )
11 r19.9rmv 3341 . . . . . . . . . . 11  |-  ( E. y  y  e.  A  ->  ( B  =  w  <->  E. z  e.  A  B  =  w )
)
1211bicomd 133 . . . . . . . . . 10  |-  ( E. y  y  e.  A  ->  ( E. z  e.  A  B  =  w  <-> 
B  =  w ) )
1310, 12sylan9bbr 444 . . . . . . . . 9  |-  ( ( E. y  y  e.  A  /\  A. x  e.  A  ( F `  x )  =  B )  ->  ( E. z  e.  A  ( F `  z )  =  w  <->  B  =  w
) )
145, 13sylan9bbr 444 . . . . . . . 8  |-  ( ( ( E. y  y  e.  A  /\  A. x  e.  A  ( F `  x )  =  B )  /\  F  Fn  A )  ->  (
w  e.  ran  F  <->  B  =  w ) )
15 velsn 3420 . . . . . . . . 9  |-  ( w  e.  { B }  <->  w  =  B )
16 eqcom 2058 . . . . . . . . 9  |-  ( w  =  B  <->  B  =  w )
1715, 16bitr2i 178 . . . . . . . 8  |-  ( B  =  w  <->  w  e.  { B } )
1814, 17syl6bb 189 . . . . . . 7  |-  ( ( ( E. y  y  e.  A  /\  A. x  e.  A  ( F `  x )  =  B )  /\  F  Fn  A )  ->  (
w  e.  ran  F  <->  w  e.  { B }
) )
1918eqrdv 2054 . . . . . 6  |-  ( ( ( E. y  y  e.  A  /\  A. x  e.  A  ( F `  x )  =  B )  /\  F  Fn  A )  ->  ran  F  =  { B }
)
2019an32s 510 . . . . 5  |-  ( ( ( E. y  y  e.  A  /\  F  Fn  A )  /\  A. x  e.  A  ( F `  x )  =  B )  ->  ran  F  =  { B }
)
2120exp31 350 . . . 4  |-  ( E. y  y  e.  A  ->  ( F  Fn  A  ->  ( A. x  e.  A  ( F `  x )  =  B  ->  ran  F  =  { B } ) ) )
2221imdistand 429 . . 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 4936 . . . 4  |-  ( F : A -onto-> { B } 
<->  ( F  Fn  A  /\  ran  F  =  { B } ) )
24 fof 5134 . . . 4  |-  ( F : A -onto-> { B }  ->  F : A --> { B } )
2523, 24sylbir 129 . . 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 135 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 101    <-> wb 102    = wceq 1259   E.wex 1397    e. wcel 1409   A.wral 2323   E.wrex 2324   {csn 3403   ran crn 4374    Fn wfn 4925   -->wf 4926   -onto->wfo 4928   ` cfv 4930
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3903  ax-pow 3955  ax-pr 3972
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-rex 2329  df-v 2576  df-sbc 2788  df-un 2950  df-in 2952  df-ss 2959  df-pw 3389  df-sn 3409  df-pr 3410  df-op 3412  df-uni 3609  df-br 3793  df-opab 3847  df-mpt 3848  df-id 4058  df-xp 4379  df-rel 4380  df-cnv 4381  df-co 4382  df-dm 4383  df-rn 4384  df-iota 4895  df-fun 4932  df-fn 4933  df-f 4934  df-fo 4936  df-fv 4938
This theorem is referenced by:  fconst3m  5408
  Copyright terms: Public domain W3C validator