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Theorem mptiniseg 5364
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 5363 . 2  |-  ( `' F " { C } )  =  {
x  e.  A  |  B  e.  { C } }
3 elsnc2g 3842 . . 3  |-  ( C  e.  V  ->  ( B  e.  { C } 
<->  B  =  C ) )
43rabbidv 2948 . 2  |-  ( C  e.  V  ->  { x  e.  A  |  B  e.  { C } }  =  { x  e.  A  |  B  =  C } )
52, 4syl5eq 2480 1  |-  ( C  e.  V  ->  ( `' F " { C } )  =  {
x  e.  A  |  B  =  C }
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1652    e. wcel 1725   {crab 2709   {csn 3814    e. cmpt 4266   `'ccnv 4877   "cima 4881
This theorem is referenced by:  ramub1lem1  13394  symgtgp  18131  csscld  19203  clsocv  19204  sqff1o  20965  dchrfi  21039  ftc1anclem6  26285  pwssplit4  27168  pwslnmlem2  27172  frlmsslss  27221
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pr 4403
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-br 4213  df-opab 4267  df-mpt 4268  df-xp 4884  df-rel 4885  df-cnv 4886  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891
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