MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  fnsnfv Structured version   Unicode version

Theorem fnsnfv 5789
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 2440 . . . 4  |-  ( y  =  ( F `  B )  <->  ( F `  B )  =  y )
2 fnbrfvb 5770 . . . 4  |-  ( ( F  Fn  A  /\  B  e.  A )  ->  ( ( F `  B )  =  y  <-> 
B F y ) )
31, 2syl5bb 250 . . 3  |-  ( ( F  Fn  A  /\  B  e.  A )  ->  ( y  =  ( F `  B )  <-> 
B F y ) )
43abbidv 2552 . 2  |-  ( ( F  Fn  A  /\  B  e.  A )  ->  { y  |  y  =  ( F `  B ) }  =  { y  |  B F y } )
5 df-sn 3822 . . 3  |-  { ( F `  B ) }  =  { y  |  y  =  ( F `  B ) }
65a1i 11 . 2  |-  ( ( F  Fn  A  /\  B  e.  A )  ->  { ( F `  B ) }  =  { y  |  y  =  ( F `  B ) } )
7 fnrel 5546 . . . 4  |-  ( F  Fn  A  ->  Rel  F )
8 relimasn 5230 . . . 4  |-  ( Rel 
F  ->  ( F " { B } )  =  { y  |  B F y } )
97, 8syl 16 . . 3  |-  ( F  Fn  A  ->  ( F " { B }
)  =  { y  |  B F y } )
109adantr 453 . 2  |-  ( ( F  Fn  A  /\  B  e.  A )  ->  ( F " { B } )  =  {
y  |  B F y } )
114, 6, 103eqtr4d 2480 1  |-  ( ( F  Fn  A  /\  B  e.  A )  ->  { ( F `  B ) }  =  ( F " { B } ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360    = wceq 1653    e. wcel 1726   {cab 2424   {csn 3816   class class class wbr 4215   "cima 4884   Rel wrel 4886    Fn wfn 5452   ` cfv 5457
This theorem is referenced by:  fnimapr  5790  funfv  5793  fvco2  5801  fvimacnvi  5847  fvimacnvALT  5852  fsn2  5911  fparlem3  6451  fparlem4  6452  domunsncan  7211  phplem4  7292  domunfican  7382  fiint  7386  infdifsn  7614  cantnfp1lem3  7639  dprdf1o  15595  cnt1  17419  xkohaus  17690  xkoptsub  17691  ustuqtop3  18278  2pthlem2  21601  eupath2lem3  21706  grpokerinj  26574  frlmlbs  27240  f1lindf  27283  funcoressn  27981
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4333  ax-nul 4341  ax-pr 4406
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-sbc 3164  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4216  df-opab 4270  df-id 4501  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-fv 5465
  Copyright terms: Public domain W3C validator