Theorem fnsnfv 5930

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.

Step	Hyp	Ref	Expression
1		eqcom 2460	. . . 4 |- ( y = ( F ` B ) <-> ( F ` B ) = y )
2		fnbrfvb 5910	. . . 4 |- ( ( F Fn A /\ B e. A ) -> ( ( F ` B ) = y <-> B F y ) )
3	1, 2	syl5bb 261	. . 3 |- ( ( F Fn A /\ B e. A ) -> ( y = ( F ` B ) <-> B F y ) )
4	3	abbidv 2571	. 2 |- ( ( F Fn A /\ B e. A ) -> { y | y = ( F ` B ) } = { y | B F y } )
5		df-sn 3971	. . 3 |- { ( F ` B ) } = { y | y = ( F ` B ) }
6	5	a1i 11	. 2 |- ( ( F Fn A /\ B e. A ) -> { ( F ` B ) } = { y | y = ( F ` B ) } )
7		fnrel 5679	. . . 4 |- ( F Fn A -> Rel F )
8		relimasn 5194	. . . 4 |- ( Rel F -> ( F " { B } ) = { y | B F y } )
9	7, 8	syl 17	. . 3 |- ( F Fn A -> ( F " { B } ) = { y | B F y } )
10	9	adantr 467	. 2 |- ( ( F Fn A /\ B e. A ) -> ( F " { B } ) = { y | B F y } )
11	4, 6, 10	3eqtr4d 2497	1 |- ( ( F Fn A /\ B e. A ) -> { ( F ` B ) } = ( F " { B } ) )

Syntax hints:   -> wi 4   /\ wa 371   = wceq 1446   e. wcel 1889   {cab 2439   {csn 3970   class class class wbr 4405   "cima 4840   Rel wrel 4842   Fn wfn 5580   ` cfv 5585

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1671  ax-4 1684  ax-5 1760  ax-6 1807  ax-7 1853  ax-9 1898  ax-10 1917  ax-11 1922  ax-12 1935  ax-13 2093  ax-ext 2433  ax-sep 4528  ax-nul 4537  ax-pr 4642

This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 988  df-tru 1449  df-ex 1666  df-nf 1670  df-sb 1800  df-eu 2305  df-mo 2306  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2583  df-ne 2626  df-ral 2744  df-rex 2745  df-rab 2748  df-v 3049  df-sbc 3270  df-dif 3409  df-un 3411  df-in 3413  df-ss 3420  df-nul 3734  df-if 3884  df-sn 3971  df-pr 3973  df-op 3977  df-uni 4202  df-br 4406  df-opab 4465  df-id 4752  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5549  df-fun 5587  df-fn 5588  df-fv 5593

This theorem is referenced by:  fnimapr  5934  funfv  5937  fvco2  5945  fvimacnvi  6001  fvimacnvALT  6006  fsn2  6067  fparlem3  6903  fparlem4  6904  suppval1  6925  suppsnop  6933  domunsncan  7677  phplem4  7759  domunfican  7849  fiint  7853  infdifsn  8167  cantnfp1lem3  8190  symgfixelsi  17088  dprdf1o  17677  frlmlbs  19367  f1lindf  19392  cnt1  20378  xkohaus  20680  xkoptsub  20681  ustuqtop3  21270  2pthlem2  25338  eupath2lem3  25719  eulerpartlemmf  29220  poimirlem4  31956  poimirlem6  31958  poimirlem7  31959  poimirlem9  31961  poimirlem13  31965  poimirlem14  31966  poimirlem16  31968  poimirlem19  31971  grpokerinj  32195  funcoressn  38638  2pthdlem2  39756