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Theorem fconst4 5957
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 5956 . 2  |-  ( F : A --> { B } 
<->  ( F  Fn  A  /\  A  C_  ( `' F " { B } ) ) )
2 cnvimass 5225 . . . . . 6  |-  ( `' F " { B } )  C_  dom  F
3 fndm 5545 . . . . . 6  |-  ( F  Fn  A  ->  dom  F  =  A )
42, 3syl5sseq 3397 . . . . 5  |-  ( F  Fn  A  ->  ( `' F " { B } )  C_  A
)
54biantrurd 496 . . . 4  |-  ( F  Fn  A  ->  ( A  C_  ( `' F " { B } )  <-> 
( ( `' F " { B } ) 
C_  A  /\  A  C_  ( `' F " { B } ) ) ) )
6 eqss 3364 . . . 4  |-  ( ( `' F " { B } )  =  A  <-> 
( ( `' F " { B } ) 
C_  A  /\  A  C_  ( `' F " { B } ) ) )
75, 6syl6bbr 256 . . 3  |-  ( F  Fn  A  ->  ( A  C_  ( `' F " { B } )  <-> 
( `' F " { B } )  =  A ) )
87pm5.32i 620 . 2  |-  ( ( F  Fn  A  /\  A  C_  ( `' F " { B } ) )  <->  ( F  Fn  A  /\  ( `' F " { B } )  =  A ) )
91, 8bitri 242 1  |-  ( F : A --> { B } 
<->  ( F  Fn  A  /\  ( `' F " { B } )  =  A ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 178    /\ wa 360    = wceq 1653    C_ wss 3321   {csn 3815   `'ccnv 4878   dom cdm 4879   "cima 4882    Fn wfn 5450   -->wf 5451
This theorem is referenced by:  lkr0f  29893
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2418  ax-sep 4331  ax-nul 4339  ax-pr 4404
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2286  df-mo 2287  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ne 2602  df-ral 2711  df-rex 2712  df-rab 2715  df-v 2959  df-sbc 3163  df-dif 3324  df-un 3326  df-in 3328  df-ss 3335  df-nul 3630  df-if 3741  df-sn 3821  df-pr 3822  df-op 3824  df-uni 4017  df-br 4214  df-opab 4268  df-mpt 4269  df-id 4499  df-xp 4885  df-rel 4886  df-cnv 4887  df-co 4888  df-dm 4889  df-rn 4890  df-res 4891  df-ima 4892  df-iota 5419  df-fun 5457  df-fn 5458  df-f 5459  df-fo 5461  df-fv 5463
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