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Theorem mptiniseg 5167
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 5166 . 2  |-  ( `' F " { C } )  =  {
x  e.  A  |  B  e.  { C } }
3 elsnc2g 3668 . . 3  |-  ( C  e.  V  ->  ( B  e.  { C } 
<->  B  =  C ) )
43rabbidv 2780 . 2  |-  ( C  e.  V  ->  { x  e.  A  |  B  e.  { C } }  =  { x  e.  A  |  B  =  C } )
52, 4syl5eq 2327 1  |-  ( C  e.  V  ->  ( `' F " { C } )  =  {
x  e.  A  |  B  =  C }
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1623    e. wcel 1684   {crab 2547   {csn 3640    e. cmpt 4077   `'ccnv 4688   "cima 4692
This theorem is referenced by:  ramub1lem1  13073  symgtgp  17784  csscld  18676  clsocv  18677  sqff1o  20420  dchrfi  20494  pwssplit4  27191  pwslnmlem2  27195  frlmsslss  27244
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-br 4024  df-opab 4078  df-mpt 4079  df-xp 4695  df-rel 4696  df-cnv 4697  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702
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