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Theorem fnsnfv 5357
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 2090 . . . 4  |-  ( y  =  ( F `  B )  <->  ( F `  B )  =  y )
2 fnbrfvb 5339 . . . 4  |-  ( ( F  Fn  A  /\  B  e.  A )  ->  ( ( F `  B )  =  y  <-> 
B F y ) )
31, 2syl5bb 190 . . 3  |-  ( ( F  Fn  A  /\  B  e.  A )  ->  ( y  =  ( F `  B )  <-> 
B F y ) )
43abbidv 2205 . 2  |-  ( ( F  Fn  A  /\  B  e.  A )  ->  { y  |  y  =  ( F `  B ) }  =  { y  |  B F y } )
5 df-sn 3450 . . 3  |-  { ( F `  B ) }  =  { y  |  y  =  ( F `  B ) }
65a1i 9 . 2  |-  ( ( F  Fn  A  /\  B  e.  A )  ->  { ( F `  B ) }  =  { y  |  y  =  ( F `  B ) } )
7 imasng 4792 . . 3  |-  ( B  e.  A  ->  ( F " { B }
)  =  { y  |  B F y } )
87adantl 271 . 2  |-  ( ( F  Fn  A  /\  B  e.  A )  ->  ( F " { B } )  =  {
y  |  B F y } )
94, 6, 83eqtr4d 2130 1  |-  ( ( F  Fn  A  /\  B  e.  A )  ->  { ( F `  B ) }  =  ( F " { B } ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    = wceq 1289    e. wcel 1438   {cab 2074   {csn 3444   class class class wbr 3843   "cima 4439    Fn wfn 5005   ` cfv 5010
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 3955  ax-pow 4007  ax-pr 4034
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-v 2621  df-sbc 2841  df-un 3003  df-in 3005  df-ss 3012  df-pw 3429  df-sn 3450  df-pr 3451  df-op 3453  df-uni 3652  df-br 3844  df-opab 3898  df-id 4118  df-xp 4442  df-rel 4443  df-cnv 4444  df-co 4445  df-dm 4446  df-rn 4447  df-res 4448  df-ima 4449  df-iota 4975  df-fun 5012  df-fn 5013  df-fv 5018
This theorem is referenced by:  fnimapr  5358  funfvdm  5361  fvco2  5367  fvimacnvi  5407  fsn2  5465  phplem4  6561  phplem4dom  6568  phplem4on  6573  fiintim  6629  fidcenumlemrks  6652  fidcenumlemr  6654  resunimafz0  10224
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