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Theorem fconst3m 5751
Description: Two ways to express a constant function. (Contributed by Jim Kingdon, 8-Jan-2019.)
Assertion
Ref Expression
fconst3m  |-  ( E. x  x  e.  A  ->  ( F : A --> { B }  <->  ( F  Fn  A  /\  A  C_  ( `' F " { B } ) ) ) )
Distinct variable group:    x, A
Allowed substitution hints:    B( x)    F( x)

Proof of Theorem fconst3m
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 fconstfvm 5750 . 2  |-  ( E. x  x  e.  A  ->  ( F : A --> { B }  <->  ( F  Fn  A  /\  A. y  e.  A  ( F `  y )  =  B ) ) )
2 fnfun 5328 . . . 4  |-  ( F  Fn  A  ->  Fun  F )
3 fndm 5330 . . . . 5  |-  ( F  Fn  A  ->  dom  F  =  A )
4 eqimss2 3225 . . . . 5  |-  ( dom 
F  =  A  ->  A  C_  dom  F )
53, 4syl 14 . . . 4  |-  ( F  Fn  A  ->  A  C_ 
dom  F )
6 funconstss 5650 . . . 4  |-  ( ( Fun  F  /\  A  C_ 
dom  F )  -> 
( A. y  e.  A  ( F `  y )  =  B  <-> 
A  C_  ( `' F " { B }
) ) )
72, 5, 6syl2anc 411 . . 3  |-  ( F  Fn  A  ->  ( A. y  e.  A  ( F `  y )  =  B  <->  A  C_  ( `' F " { B } ) ) )
87pm5.32i 454 . 2  |-  ( ( F  Fn  A  /\  A. y  e.  A  ( F `  y )  =  B )  <->  ( F  Fn  A  /\  A  C_  ( `' F " { B } ) ) )
91, 8bitrdi 196 1  |-  ( E. x  x  e.  A  ->  ( F : A --> { B }  <->  ( F  Fn  A  /\  A  C_  ( `' F " { B } ) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    <-> wb 105    = wceq 1364   E.wex 1503    e. wcel 2160   A.wral 2468    C_ wss 3144   {csn 3607   `'ccnv 4640   dom cdm 4641   "cima 4644   Fun wfun 5225    Fn wfn 5226   -->wf 5227   ` cfv 5231
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4189  ax-pr 4224
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-v 2754  df-sbc 2978  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-br 4019  df-opab 4080  df-mpt 4081  df-id 4308  df-xp 4647  df-rel 4648  df-cnv 4649  df-co 4650  df-dm 4651  df-rn 4652  df-res 4653  df-ima 4654  df-iota 5193  df-fun 5233  df-fn 5234  df-f 5235  df-fo 5237  df-fv 5239
This theorem is referenced by:  fconst4m  5752
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