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Theorem fconst3m 5530
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 5529 . 2  |-  ( E. x  x  e.  A  ->  ( F : A --> { B }  <->  ( F  Fn  A  /\  A. y  e.  A  ( F `  y )  =  B ) ) )
2 fnfun 5124 . . . 4  |-  ( F  Fn  A  ->  Fun  F )
3 fndm 5126 . . . . 5  |-  ( F  Fn  A  ->  dom  F  =  A )
4 eqimss2 3080 . . . . 5  |-  ( dom 
F  =  A  ->  A  C_  dom  F )
53, 4syl 14 . . . 4  |-  ( F  Fn  A  ->  A  C_ 
dom  F )
6 funconstss 5431 . . . 4  |-  ( ( Fun  F  /\  A  C_ 
dom  F )  -> 
( A. y  e.  A  ( F `  y )  =  B  <-> 
A  C_  ( `' F " { B }
) ) )
72, 5, 6syl2anc 404 . . 3  |-  ( F  Fn  A  ->  ( A. y  e.  A  ( F `  y )  =  B  <->  A  C_  ( `' F " { B } ) ) )
87pm5.32i 443 . 2  |-  ( ( F  Fn  A  /\  A. y  e.  A  ( F `  y )  =  B )  <->  ( F  Fn  A  /\  A  C_  ( `' F " { B } ) ) )
91, 8syl6bb 195 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 103    <-> wb 104    = wceq 1290   E.wex 1427    e. wcel 1439   A.wral 2360    C_ wss 3000   {csn 3450   `'ccnv 4450   dom cdm 4451   "cima 4454   Fun wfun 5022    Fn wfn 5023   -->wf 5024   ` cfv 5028
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-14 1451  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071  ax-sep 3963  ax-pow 4015  ax-pr 4045
This theorem depends on definitions:  df-bi 116  df-3an 927  df-tru 1293  df-nf 1396  df-sb 1694  df-eu 1952  df-mo 1953  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-ral 2365  df-rex 2366  df-v 2622  df-sbc 2842  df-un 3004  df-in 3006  df-ss 3013  df-pw 3435  df-sn 3456  df-pr 3457  df-op 3459  df-uni 3660  df-br 3852  df-opab 3906  df-mpt 3907  df-id 4129  df-xp 4457  df-rel 4458  df-cnv 4459  df-co 4460  df-dm 4461  df-rn 4462  df-res 4463  df-ima 4464  df-iota 4993  df-fun 5030  df-fn 5031  df-f 5032  df-fo 5034  df-fv 5036
This theorem is referenced by:  fconst4m  5531
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