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Theorem mptiniseg 5330
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 5329 . 2  |-  ( `' F " { C } )  =  {
x  e.  A  |  B  e.  { C } }
3 elsnc2g 3905 . . 3  |-  ( C  e.  V  ->  ( B  e.  { C } 
<->  B  =  C ) )
43rabbidv 2962 . 2  |-  ( C  e.  V  ->  { x  e.  A  |  B  e.  { C } }  =  { x  e.  A  |  B  =  C } )
52, 4syl5eq 2485 1  |-  ( C  e.  V  ->  ( `' F " { C } )  =  {
x  e.  A  |  B  =  C }
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1369    e. wcel 1756   {crab 2717   {csn 3875    e. cmpt 4348   `'ccnv 4837   "cima 4841
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2422  ax-sep 4411  ax-nul 4419  ax-pr 4529
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-ral 2718  df-rex 2719  df-rab 2722  df-v 2972  df-dif 3329  df-un 3331  df-in 3333  df-ss 3340  df-nul 3636  df-if 3790  df-sn 3876  df-pr 3878  df-op 3882  df-br 4291  df-opab 4349  df-mpt 4350  df-xp 4844  df-rel 4845  df-cnv 4846  df-dm 4848  df-rn 4849  df-res 4850  df-ima 4851
This theorem is referenced by:  ramub1lem1  14085  frlmsslss  18196  symgtgp  19670  csscld  20759  clsocv  20760  sqff1o  22518  dchrfi  22592  dvtanlem  28438  ftc1anclem6  28469  pwssplit4  29439  pwslnmlem2  29443
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