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Theorem fconst4 6143
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 6142 . 2  |-  ( F : A --> { B } 
<->  ( F  Fn  A  /\  A  C_  ( `' F " { B } ) ) )
2 cnvimass 5206 . . . . . 6  |-  ( `' F " { B } )  C_  dom  F
3 fndm 5692 . . . . . 6  |-  ( F  Fn  A  ->  dom  F  =  A )
42, 3syl5sseq 3514 . . . . 5  |-  ( F  Fn  A  ->  ( `' F " { B } )  C_  A
)
54biantrurd 511 . . . 4  |-  ( F  Fn  A  ->  ( A  C_  ( `' F " { B } )  <-> 
( ( `' F " { B } ) 
C_  A  /\  A  C_  ( `' F " { B } ) ) ) )
6 eqss 3481 . . . 4  |-  ( ( `' F " { B } )  =  A  <-> 
( ( `' F " { B } ) 
C_  A  /\  A  C_  ( `' F " { B } ) ) )
75, 6syl6bbr 267 . . 3  |-  ( F  Fn  A  ->  ( A  C_  ( `' F " { B } )  <-> 
( `' F " { B } )  =  A ) )
87pm5.32i 642 . 2  |-  ( ( F  Fn  A  /\  A  C_  ( `' F " { B } ) )  <->  ( F  Fn  A  /\  ( `' F " { B } )  =  A ) )
91, 8bitri 253 1  |-  ( F : A --> { B } 
<->  ( F  Fn  A  /\  ( `' F " { B } )  =  A ) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 188    /\ wa 371    = wceq 1438    C_ wss 3438   {csn 3998   `'ccnv 4851   dom cdm 4852   "cima 4855    Fn wfn 5595   -->wf 5596
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1666  ax-4 1679  ax-5 1749  ax-6 1795  ax-7 1840  ax-9 1873  ax-10 1888  ax-11 1893  ax-12 1906  ax-13 2054  ax-ext 2401  ax-sep 4545  ax-nul 4554  ax-pr 4659
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 985  df-tru 1441  df-ex 1661  df-nf 1665  df-sb 1788  df-eu 2270  df-mo 2271  df-clab 2409  df-cleq 2415  df-clel 2418  df-nfc 2573  df-ne 2621  df-ral 2781  df-rex 2782  df-rab 2785  df-v 3084  df-sbc 3302  df-dif 3441  df-un 3443  df-in 3445  df-ss 3452  df-nul 3764  df-if 3912  df-sn 3999  df-pr 4001  df-op 4005  df-uni 4219  df-br 4423  df-opab 4482  df-mpt 4483  df-id 4767  df-xp 4858  df-rel 4859  df-cnv 4860  df-co 4861  df-dm 4862  df-rn 4863  df-res 4864  df-ima 4865  df-iota 5564  df-fun 5602  df-fn 5603  df-f 5604  df-fv 5608
This theorem is referenced by:  lkr0f  32627
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