MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  fconst3 Structured version   Unicode version

Theorem fconst3 6118
Description: Two ways to express a constant function. (Contributed by NM, 15-Mar-2007.)
Assertion
Ref Expression
fconst3  |-  ( F : A --> { B } 
<->  ( F  Fn  A  /\  A  C_  ( `' F " { B } ) ) )

Proof of Theorem fconst3
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 fconstfv 6116 . 2  |-  ( F : A --> { B } 
<->  ( F  Fn  A  /\  A. x  e.  A  ( F `  x )  =  B ) )
2 fnfun 5661 . . . 4  |-  ( F  Fn  A  ->  Fun  F )
3 fndm 5663 . . . . 5  |-  ( F  Fn  A  ->  dom  F  =  A )
4 eqimss2 3497 . . . . 5  |-  ( dom 
F  =  A  ->  A  C_  dom  F )
53, 4syl 17 . . . 4  |-  ( F  Fn  A  ->  A  C_ 
dom  F )
6 funconstss 5985 . . . 4  |-  ( ( Fun  F  /\  A  C_ 
dom  F )  -> 
( A. x  e.  A  ( F `  x )  =  B  <-> 
A  C_  ( `' F " { B }
) ) )
72, 5, 6syl2anc 661 . . 3  |-  ( F  Fn  A  ->  ( A. x  e.  A  ( F `  x )  =  B  <->  A  C_  ( `' F " { B } ) ) )
87pm5.32i 637 . 2  |-  ( ( F  Fn  A  /\  A. x  e.  A  ( F `  x )  =  B )  <->  ( F  Fn  A  /\  A  C_  ( `' F " { B } ) ) )
91, 8bitri 251 1  |-  ( F : A --> { B } 
<->  ( F  Fn  A  /\  A  C_  ( `' F " { B } ) ) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 186    /\ wa 369    = wceq 1407   A.wral 2756    C_ wss 3416   {csn 3974   `'ccnv 4824   dom cdm 4825   "cima 4828   Fun wfun 5565    Fn wfn 5566   -->wf 5567   ` cfv 5571
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1641  ax-4 1654  ax-5 1727  ax-6 1773  ax-7 1816  ax-9 1848  ax-10 1863  ax-11 1868  ax-12 1880  ax-13 2028  ax-ext 2382  ax-sep 4519  ax-nul 4527  ax-pr 4632
This theorem depends on definitions:  df-bi 187  df-or 370  df-an 371  df-3an 978  df-tru 1410  df-ex 1636  df-nf 1640  df-sb 1766  df-eu 2244  df-mo 2245  df-clab 2390  df-cleq 2396  df-clel 2399  df-nfc 2554  df-ne 2602  df-ral 2761  df-rex 2762  df-rab 2765  df-v 3063  df-sbc 3280  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-nul 3741  df-if 3888  df-sn 3975  df-pr 3977  df-op 3981  df-uni 4194  df-br 4398  df-opab 4456  df-mpt 4457  df-id 4740  df-xp 4831  df-rel 4832  df-cnv 4833  df-co 4834  df-dm 4835  df-rn 4836  df-res 4837  df-ima 4838  df-iota 5535  df-fun 5573  df-fn 5574  df-f 5575  df-fv 5579
This theorem is referenced by:  fconst4  6119  dnsconst  20174
  Copyright terms: Public domain W3C validator