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Theorem fconst3m 5607
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 5606 . 2  |-  ( E. x  x  e.  A  ->  ( F : A --> { B }  <->  ( F  Fn  A  /\  A. y  e.  A  ( F `  y )  =  B ) ) )
2 fnfun 5190 . . . 4  |-  ( F  Fn  A  ->  Fun  F )
3 fndm 5192 . . . . 5  |-  ( F  Fn  A  ->  dom  F  =  A )
4 eqimss2 3122 . . . . 5  |-  ( dom 
F  =  A  ->  A  C_  dom  F )
53, 4syl 14 . . . 4  |-  ( F  Fn  A  ->  A  C_ 
dom  F )
6 funconstss 5506 . . . 4  |-  ( ( Fun  F  /\  A  C_ 
dom  F )  -> 
( A. y  e.  A  ( F `  y )  =  B  <-> 
A  C_  ( `' F " { B }
) ) )
72, 5, 6syl2anc 408 . . 3  |-  ( F  Fn  A  ->  ( A. y  e.  A  ( F `  y )  =  B  <->  A  C_  ( `' F " { B } ) ) )
87pm5.32i 449 . 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 1316   E.wex 1453    e. wcel 1465   A.wral 2393    C_ wss 3041   {csn 3497   `'ccnv 4508   dom cdm 4509   "cima 4512   Fun wfun 5087    Fn wfn 5088   -->wf 5089   ` cfv 5093
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016  ax-pow 4068  ax-pr 4101
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ral 2398  df-rex 2399  df-v 2662  df-sbc 2883  df-un 3045  df-in 3047  df-ss 3054  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-uni 3707  df-br 3900  df-opab 3960  df-mpt 3961  df-id 4185  df-xp 4515  df-rel 4516  df-cnv 4517  df-co 4518  df-dm 4519  df-rn 4520  df-res 4521  df-ima 4522  df-iota 5058  df-fun 5095  df-fn 5096  df-f 5097  df-fo 5099  df-fv 5101
This theorem is referenced by:  fconst4m  5608
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