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Theorem mptiniseg 4925
Description: Converse singleton image of a function defined by maps-to. (Contributed by Stefan O'Rear, 25-Jan-2015.)
Hypothesis
Ref Expression
dmmpt2.1  |-  F  =  ( x  e.  A  |->  B )
Assertion
Ref Expression
mptiniseg  |-  ( C  e.  V  ->  ( `' F " { C } )  =  {
x  e.  A  |  B  =  C }
)
Distinct variable groups:    x, C    x, V
Allowed substitution hints:    A( x)    B( x)    F( x)

Proof of Theorem mptiniseg
StepHypRef Expression
1 dmmpt2.1 . . 3  |-  F  =  ( x  e.  A  |->  B )
21mptpreima 4924 . 2  |-  ( `' F " { C } )  =  {
x  e.  A  |  B  e.  { C } }
3 elsn2g 3477 . . 3  |-  ( C  e.  V  ->  ( B  e.  { C } 
<->  B  =  C ) )
43rabbidv 2608 . 2  |-  ( C  e.  V  ->  { x  e.  A  |  B  e.  { C } }  =  { x  e.  A  |  B  =  C } )
52, 4syl5eq 2132 1  |-  ( C  e.  V  ->  ( `' F " { C } )  =  {
x  e.  A  |  B  =  C }
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1289    e. wcel 1438   {crab 2363   {csn 3446    |-> cmpt 3899   `'ccnv 4437   "cima 4441
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 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3957  ax-pow 4009  ax-pr 4036
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-rab 2368  df-v 2621  df-un 3003  df-in 3005  df-ss 3012  df-pw 3431  df-sn 3452  df-pr 3453  df-op 3455  df-br 3846  df-opab 3900  df-mpt 3901  df-xp 4444  df-rel 4445  df-cnv 4446  df-dm 4448  df-rn 4449  df-res 4450  df-ima 4451
This theorem is referenced by: (None)
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