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Theorem fconst3 6117
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 6116 . 2  |-  ( F : A --> { B } 
<->  ( F  Fn  A  /\  A. x  e.  A  ( F `  x )  =  B ) )
2 fnfun 5671 . . . 4  |-  ( F  Fn  A  ->  Fun  F )
3 fndm 5673 . . . . 5  |-  ( F  Fn  A  ->  dom  F  =  A )
4 eqimss2 3552 . . . . 5  |-  ( dom 
F  =  A  ->  A  C_  dom  F )
53, 4syl 16 . . . 4  |-  ( F  Fn  A  ->  A  C_ 
dom  F )
6 funconstss 5992 . . . 4  |-  ( ( Fun  F  /\  A  C_ 
dom  F )  -> 
( A. x  e.  A  ( F `  x )  =  B  <-> 
A  C_  ( `' F " { B }
) ) )
72, 5, 6syl2anc 661 . . 3  |-  ( F  Fn  A  ->  ( A. x  e.  A  ( F `  x )  =  B  <->  A  C_  ( `' F " { B } ) ) )
87pm5.32i 637 . 2  |-  ( ( F  Fn  A  /\  A. x  e.  A  ( F `  x )  =  B )  <->  ( F  Fn  A  /\  A  C_  ( `' F " { B } ) ) )
91, 8bitri 249 1  |-  ( F : A --> { B } 
<->  ( F  Fn  A  /\  A  C_  ( `' F " { B } ) ) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    /\ wa 369    = wceq 1374   A.wral 2809    C_ wss 3471   {csn 4022   `'ccnv 4993   dom cdm 4994   "cima 4997   Fun wfun 5575    Fn wfn 5576   -->wf 5577   ` cfv 5581
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1963  ax-ext 2440  ax-sep 4563  ax-nul 4571  ax-pr 4681
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2274  df-mo 2275  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2612  df-ne 2659  df-ral 2814  df-rex 2815  df-rab 2818  df-v 3110  df-sbc 3327  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3781  df-if 3935  df-sn 4023  df-pr 4025  df-op 4029  df-uni 4241  df-br 4443  df-opab 4501  df-mpt 4502  df-id 4790  df-xp 5000  df-rel 5001  df-cnv 5002  df-co 5003  df-dm 5004  df-rn 5005  df-res 5006  df-ima 5007  df-iota 5544  df-fun 5583  df-fn 5584  df-f 5585  df-fo 5587  df-fv 5589
This theorem is referenced by:  fconst4  6118  dnsconst  19640
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