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Theorem fnsnfv 5232
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 2042 . . . 4  |-  ( y  =  ( F `  B )  <->  ( F `  B )  =  y )
2 fnbrfvb 5214 . . . 4  |-  ( ( F  Fn  A  /\  B  e.  A )  ->  ( ( F `  B )  =  y  <-> 
B F y ) )
31, 2syl5bb 181 . . 3  |-  ( ( F  Fn  A  /\  B  e.  A )  ->  ( y  =  ( F `  B )  <-> 
B F y ) )
43abbidv 2155 . 2  |-  ( ( F  Fn  A  /\  B  e.  A )  ->  { y  |  y  =  ( F `  B ) }  =  { y  |  B F y } )
5 df-sn 3381 . . 3  |-  { ( F `  B ) }  =  { y  |  y  =  ( F `  B ) }
65a1i 9 . 2  |-  ( ( F  Fn  A  /\  B  e.  A )  ->  { ( F `  B ) }  =  { y  |  y  =  ( F `  B ) } )
7 imasng 4690 . . 3  |-  ( B  e.  A  ->  ( F " { B }
)  =  { y  |  B F y } )
87adantl 262 . 2  |-  ( ( F  Fn  A  /\  B  e.  A )  ->  ( F " { B } )  =  {
y  |  B F y } )
94, 6, 83eqtr4d 2082 1  |-  ( ( F  Fn  A  /\  B  e.  A )  ->  { ( F `  B ) }  =  ( F " { B } ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 97    = wceq 1243    e. wcel 1393   {cab 2026   {csn 3375   class class class wbr 3764   "cima 4348    Fn wfn 4897   ` cfv 4902
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944
This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-sbc 2765  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-br 3765  df-opab 3819  df-id 4030  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-res 4357  df-ima 4358  df-iota 4867  df-fun 4904  df-fn 4905  df-fv 4910
This theorem is referenced by:  fnimapr  5233  funfvdm  5236  fvco2  5242  fvimacnvi  5281  fsn2  5337  phplem4  6318  phplem4dom  6324  phplem4on  6329
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