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Theorem fconst3 5735
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 5734 . 2  |-  ( F : A --> { B } 
<->  ( F  Fn  A  /\  A. x  e.  A  ( F `  x )  =  B ) )
2 fnfun 5341 . . . 4  |-  ( F  Fn  A  ->  Fun  F )
3 fndm 5343 . . . . 5  |-  ( F  Fn  A  ->  dom  F  =  A )
4 eqimss2 3231 . . . . 5  |-  ( dom 
F  =  A  ->  A  C_  dom  F )
53, 4syl 15 . . . 4  |-  ( F  Fn  A  ->  A  C_ 
dom  F )
6 funconstss 5643 . . . 4  |-  ( ( Fun  F  /\  A  C_ 
dom  F )  -> 
( A. x  e.  A  ( F `  x )  =  B  <-> 
A  C_  ( `' F " { B }
) ) )
72, 5, 6syl2anc 642 . . 3  |-  ( F  Fn  A  ->  ( A. x  e.  A  ( F `  x )  =  B  <->  A  C_  ( `' F " { B } ) ) )
87pm5.32i 618 . 2  |-  ( ( F  Fn  A  /\  A. x  e.  A  ( F `  x )  =  B )  <->  ( F  Fn  A  /\  A  C_  ( `' F " { B } ) ) )
91, 8bitri 240 1  |-  ( F : A --> { B } 
<->  ( F  Fn  A  /\  A  C_  ( `' F " { B } ) ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    /\ wa 358    = wceq 1623   A.wral 2543    C_ wss 3152   {csn 3640   `'ccnv 4688   dom cdm 4689   "cima 4692   Fun wfun 5249    Fn wfn 5250   -->wf 5251   ` cfv 5255
This theorem is referenced by:  fconst4  5736  dnsconst  17106
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-fo 5261  df-fv 5263
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