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Theorem fconst3m 5639
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 5638 . 2  |-  ( E. x  x  e.  A  ->  ( F : A --> { B }  <->  ( F  Fn  A  /\  A. y  e.  A  ( F `  y )  =  B ) ) )
2 fnfun 5220 . . . 4  |-  ( F  Fn  A  ->  Fun  F )
3 fndm 5222 . . . . 5  |-  ( F  Fn  A  ->  dom  F  =  A )
4 eqimss2 3152 . . . . 5  |-  ( dom 
F  =  A  ->  A  C_  dom  F )
53, 4syl 14 . . . 4  |-  ( F  Fn  A  ->  A  C_ 
dom  F )
6 funconstss 5538 . . . 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 1331   E.wex 1468    e. wcel 1480   A.wral 2416    C_ wss 3071   {csn 3527   `'ccnv 4538   dom cdm 4539   "cima 4542   Fun wfun 5117    Fn wfn 5118   -->wf 5119   ` cfv 5123
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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-v 2688  df-sbc 2910  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-br 3930  df-opab 3990  df-mpt 3991  df-id 4215  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-fo 5129  df-fv 5131
This theorem is referenced by:  fconst4m  5640
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