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Theorem fconst4 5937
Description: Two ways to express a constant function. (Contributed by NM, 8-Mar-2007.)
Assertion
Ref Expression
fconst4  |-  ( F : A --> { B } 
<->  ( F  Fn  A  /\  ( `' F " { B } )  =  A ) )

Proof of Theorem fconst4
StepHypRef Expression
1 fconst3 5936 . 2  |-  ( F : A --> { B } 
<->  ( F  Fn  A  /\  A  C_  ( `' F " { B } ) ) )
2 cnvimass 5184 . . . . . 6  |-  ( `' F " { B } )  C_  dom  F
3 fndm 5505 . . . . . 6  |-  ( F  Fn  A  ->  dom  F  =  A )
42, 3syl5sseq 3399 . . . . 5  |-  ( F  Fn  A  ->  ( `' F " { B } )  C_  A
)
54biantrurd 508 . . . 4  |-  ( F  Fn  A  ->  ( A  C_  ( `' F " { B } )  <-> 
( ( `' F " { B } ) 
C_  A  /\  A  C_  ( `' F " { B } ) ) ) )
6 eqss 3366 . . . 4  |-  ( ( `' F " { B } )  =  A  <-> 
( ( `' F " { B } ) 
C_  A  /\  A  C_  ( `' F " { B } ) ) )
75, 6syl6bbr 263 . . 3  |-  ( F  Fn  A  ->  ( A  C_  ( `' F " { B } )  <-> 
( `' F " { B } )  =  A ) )
87pm5.32i 637 . 2  |-  ( ( F  Fn  A  /\  A  C_  ( `' F " { B } ) )  <->  ( F  Fn  A  /\  ( `' F " { B } )  =  A ) )
91, 8bitri 249 1  |-  ( F : A --> { B } 
<->  ( F  Fn  A  /\  ( `' F " { B } )  =  A ) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    /\ wa 369    = wceq 1369    C_ wss 3323   {csn 3872   `'ccnv 4834   dom cdm 4835   "cima 4838    Fn wfn 5408   -->wf 5409
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419  ax-sep 4408  ax-nul 4416  ax-pr 4526
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2715  df-rex 2716  df-rab 2719  df-v 2969  df-sbc 3182  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-nul 3633  df-if 3787  df-sn 3873  df-pr 3875  df-op 3879  df-uni 4087  df-br 4288  df-opab 4346  df-mpt 4347  df-id 4631  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-rn 4846  df-res 4847  df-ima 4848  df-iota 5376  df-fun 5415  df-fn 5416  df-f 5417  df-fo 5419  df-fv 5421
This theorem is referenced by:  lkr0f  32579
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