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Theorem fconst4m 5413
Description: Two ways to express a constant function. (Contributed by NM, 8-Mar-2007.)
Assertion
Ref Expression
fconst4m  |-  ( E. x  x  e.  A  ->  ( F : A --> { B }  <->  ( F  Fn  A  /\  ( `' F " { B } )  =  A ) ) )
Distinct variable group:    x, A
Allowed substitution hints:    B( x)    F( x)

Proof of Theorem fconst4m
StepHypRef Expression
1 fconst3m 5412 . 2  |-  ( E. x  x  e.  A  ->  ( F : A --> { B }  <->  ( F  Fn  A  /\  A  C_  ( `' F " { B } ) ) ) )
2 cnvimass 4718 . . . . . 6  |-  ( `' F " { B } )  C_  dom  F
3 fndm 5029 . . . . . 6  |-  ( F  Fn  A  ->  dom  F  =  A )
42, 3syl5sseq 3048 . . . . 5  |-  ( F  Fn  A  ->  ( `' F " { B } )  C_  A
)
54biantrurd 299 . . . 4  |-  ( F  Fn  A  ->  ( A  C_  ( `' F " { B } )  <-> 
( ( `' F " { B } ) 
C_  A  /\  A  C_  ( `' F " { B } ) ) ) )
6 eqss 3015 . . . 4  |-  ( ( `' F " { B } )  =  A  <-> 
( ( `' F " { B } ) 
C_  A  /\  A  C_  ( `' F " { B } ) ) )
75, 6syl6bbr 196 . . 3  |-  ( F  Fn  A  ->  ( A  C_  ( `' F " { B } )  <-> 
( `' F " { B } )  =  A ) )
87pm5.32i 442 . 2  |-  ( ( F  Fn  A  /\  A  C_  ( `' F " { B } ) )  <->  ( F  Fn  A  /\  ( `' F " { B } )  =  A ) )
91, 8syl6bb 194 1  |-  ( E. x  x  e.  A  ->  ( F : A --> { B }  <->  ( F  Fn  A  /\  ( `' F " { B } )  =  A ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    <-> wb 103    = wceq 1285   E.wex 1422    e. wcel 1434    C_ wss 2974   {csn 3406   `'ccnv 4370   dom cdm 4371   "cima 4374    Fn wfn 4927   -->wf 4928
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-sep 3904  ax-pow 3956  ax-pr 3972
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1687  df-eu 1945  df-mo 1946  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ral 2354  df-rex 2355  df-v 2604  df-sbc 2817  df-un 2978  df-in 2980  df-ss 2987  df-pw 3392  df-sn 3412  df-pr 3413  df-op 3415  df-uni 3610  df-br 3794  df-opab 3848  df-mpt 3849  df-id 4056  df-xp 4377  df-rel 4378  df-cnv 4379  df-co 4380  df-dm 4381  df-rn 4382  df-res 4383  df-ima 4384  df-iota 4897  df-fun 4934  df-fn 4935  df-f 4936  df-fo 4938  df-fv 4940
This theorem is referenced by: (None)
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