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Theorem fnsnfv 5662
Description: Singleton of function value. (Contributed by NM, 22-May-1998.)
Assertion
Ref Expression
fnsnfv  |-  ( ( F  Fn  A  /\  B  e.  A )  ->  { ( F `  B ) }  =  ( F " { B } ) )

Proof of Theorem fnsnfv
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 eqcom 2360 . . . 4  |-  ( y  =  ( F `  B )  <->  ( F `  B )  =  y )
2 fnbrfvb 5643 . . . 4  |-  ( ( F  Fn  A  /\  B  e.  A )  ->  ( ( F `  B )  =  y  <-> 
B F y ) )
31, 2syl5bb 248 . . 3  |-  ( ( F  Fn  A  /\  B  e.  A )  ->  ( y  =  ( F `  B )  <-> 
B F y ) )
43abbidv 2472 . 2  |-  ( ( F  Fn  A  /\  B  e.  A )  ->  { y  |  y  =  ( F `  B ) }  =  { y  |  B F y } )
5 df-sn 3722 . . 3  |-  { ( F `  B ) }  =  { y  |  y  =  ( F `  B ) }
65a1i 10 . 2  |-  ( ( F  Fn  A  /\  B  e.  A )  ->  { ( F `  B ) }  =  { y  |  y  =  ( F `  B ) } )
7 fnrel 5421 . . . 4  |-  ( F  Fn  A  ->  Rel  F )
8 relimasn 5115 . . . 4  |-  ( Rel 
F  ->  ( F " { B } )  =  { y  |  B F y } )
97, 8syl 15 . . 3  |-  ( F  Fn  A  ->  ( F " { B }
)  =  { y  |  B F y } )
109adantr 451 . 2  |-  ( ( F  Fn  A  /\  B  e.  A )  ->  ( F " { B } )  =  {
y  |  B F y } )
114, 6, 103eqtr4d 2400 1  |-  ( ( F  Fn  A  /\  B  e.  A )  ->  { ( F `  B ) }  =  ( F " { B } ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1642    e. wcel 1710   {cab 2344   {csn 3716   class class class wbr 4102   "cima 4771   Rel wrel 4773    Fn wfn 5329   ` cfv 5334
This theorem is referenced by:  fnimapr  5663  funfv  5666  fvco2  5674  fvimacnvi  5719  fvimacnvALT  5724  fsn2  5778  fparlem3  6304  fparlem4  6305  domunsncan  7047  phplem4  7128  domunfican  7216  fiint  7220  infdifsn  7444  cantnfp1lem3  7469  dprdf1o  15360  cnt1  17178  xkohaus  17447  xkoptsub  17448  ustuqtop3  23547  eupath2lem3  24307  grpokerinj  25898  frlmlbs  26572  f1lindf  26615  funcoressn  27315  2pthonlem2  27719
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-sep 4220  ax-nul 4228  ax-pr 4293
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-ral 2624  df-rex 2625  df-rab 2628  df-v 2866  df-sbc 3068  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-nul 3532  df-if 3642  df-sn 3722  df-pr 3723  df-op 3725  df-uni 3907  df-br 4103  df-opab 4157  df-id 4388  df-xp 4774  df-rel 4775  df-cnv 4776  df-co 4777  df-dm 4778  df-rn 4779  df-res 4780  df-ima 4781  df-iota 5298  df-fun 5336  df-fn 5337  df-fv 5342
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