ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  fconst3m Unicode version

Theorem fconst3m 5406
Description: Two ways to express a constant function. (Contributed by Jim Kingdon, 8-Jan-2019.)
Assertion
Ref Expression
fconst3m  |-  ( E. x  x  e.  A  ->  ( F : A --> { B }  <->  ( F  Fn  A  /\  A  C_  ( `' F " { B } ) ) ) )
Distinct variable group:    x, A
Allowed substitution hints:    B( x)    F( x)

Proof of Theorem fconst3m
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 fconstfvm 5405 . 2  |-  ( E. x  x  e.  A  ->  ( F : A --> { B }  <->  ( F  Fn  A  /\  A. y  e.  A  ( F `  y )  =  B ) ) )
2 fnfun 5021 . . . 4  |-  ( F  Fn  A  ->  Fun  F )
3 fndm 5023 . . . . 5  |-  ( F  Fn  A  ->  dom  F  =  A )
4 eqimss2 3053 . . . . 5  |-  ( dom 
F  =  A  ->  A  C_  dom  F )
53, 4syl 14 . . . 4  |-  ( F  Fn  A  ->  A  C_ 
dom  F )
6 funconstss 5311 . . . 4  |-  ( ( Fun  F  /\  A  C_ 
dom  F )  -> 
( A. y  e.  A  ( F `  y )  =  B  <-> 
A  C_  ( `' F " { B }
) ) )
72, 5, 6syl2anc 403 . . 3  |-  ( F  Fn  A  ->  ( A. y  e.  A  ( F `  y )  =  B  <->  A  C_  ( `' F " { B } ) ) )
87pm5.32i 442 . 2  |-  ( ( F  Fn  A  /\  A. y  e.  A  ( F `  y )  =  B )  <->  ( F  Fn  A  /\  A  C_  ( `' F " { B } ) ) )
91, 8syl6bb 194 1  |-  ( E. x  x  e.  A  ->  ( F : A --> { B }  <->  ( F  Fn  A  /\  A  C_  ( `' F " { B } ) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    <-> wb 103    = wceq 1285   E.wex 1422    e. wcel 1434   A.wral 2349    C_ wss 2974   {csn 3400   `'ccnv 4364   dom cdm 4365   "cima 4368   Fun wfun 4920    Fn wfn 4921   -->wf 4922   ` cfv 4926
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 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-sep 3898  ax-pow 3950  ax-pr 3966
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1687  df-eu 1945  df-mo 1946  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ral 2354  df-rex 2355  df-v 2604  df-sbc 2817  df-un 2978  df-in 2980  df-ss 2987  df-pw 3386  df-sn 3406  df-pr 3407  df-op 3409  df-uni 3604  df-br 3788  df-opab 3842  df-mpt 3843  df-id 4050  df-xp 4371  df-rel 4372  df-cnv 4373  df-co 4374  df-dm 4375  df-rn 4376  df-res 4377  df-ima 4378  df-iota 4891  df-fun 4928  df-fn 4929  df-f 4930  df-fo 4932  df-fv 4934
This theorem is referenced by:  fconst4m  5407
  Copyright terms: Public domain W3C validator