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Theorem fconst4 5736
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 5735 . 2  |-  ( F : A --> { B } 
<->  ( F  Fn  A  /\  A  C_  ( `' F " { B } ) ) )
2 cnvimass 5033 . . . . . 6  |-  ( `' F " { B } )  C_  dom  F
3 fndm 5343 . . . . . 6  |-  ( F  Fn  A  ->  dom  F  =  A )
42, 3syl5sseq 3226 . . . . 5  |-  ( F  Fn  A  ->  ( `' F " { B } )  C_  A
)
54biantrurd 494 . . . 4  |-  ( F  Fn  A  ->  ( A  C_  ( `' F " { B } )  <-> 
( ( `' F " { B } ) 
C_  A  /\  A  C_  ( `' F " { B } ) ) ) )
6 eqss 3194 . . . 4  |-  ( ( `' F " { B } )  =  A  <-> 
( ( `' F " { B } ) 
C_  A  /\  A  C_  ( `' F " { B } ) ) )
75, 6syl6bbr 254 . . 3  |-  ( F  Fn  A  ->  ( A  C_  ( `' F " { B } )  <-> 
( `' F " { B } )  =  A ) )
87pm5.32i 618 . 2  |-  ( ( F  Fn  A  /\  A  C_  ( `' F " { B } ) )  <->  ( F  Fn  A  /\  ( `' F " { B } )  =  A ) )
91, 8bitri 240 1  |-  ( F : A --> { B } 
<->  ( F  Fn  A  /\  ( `' F " { B } )  =  A ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    /\ wa 358    = wceq 1623    C_ wss 3152   {csn 3640   `'ccnv 4688   dom cdm 4689   "cima 4692    Fn wfn 5250   -->wf 5251
This theorem is referenced by:  lkr0f  29284
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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