Theorem fnsnfv 5739

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 2435	. . . 4 |- ( y = ( F ` B ) <-> ( F ` B ) = y )
2		fnbrfvb 5720	. . . 4 |- ( ( F Fn A /\ B e. A ) -> ( ( F ` B ) = y <-> B F y ) )
3	1, 2	syl5bb 257	. . 3 |- ( ( F Fn A /\ B e. A ) -> ( y = ( F ` B ) <-> B F y ) )
4	3	abbidv 2547	. 2 |- ( ( F Fn A /\ B e. A ) -> { y | y = ( F ` B ) } = { y | B F y } )
5		df-sn 3866	. . 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 5497	. . . 4 |- ( F Fn A -> Rel F )
8		relimasn 5180	. . . 4 |- ( Rel F -> ( F " { B } ) = { y | B F y } )
9	7, 8	syl 16	. . 3 |- ( F Fn A -> ( F " { B } ) = { y | B F y } )
10	9	adantr 462	. 2 |- ( ( F Fn A /\ B e. A ) -> ( F " { B } ) = { y | B F y } )
11	4, 6, 10	3eqtr4d 2475	1 |- ( ( F Fn A /\ B e. A ) -> { ( F ` B ) } = ( F " { B } ) )

Syntax hints:   -> wi 4   /\ wa 369   = wceq 1362   e. wcel 1755   {cab 2419   {csn 3865   class class class wbr 4280   "cima 4830   Rel wrel 4832   Fn wfn 5401   ` cfv 5406

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1594  ax-4 1605  ax-5 1669  ax-6 1707  ax-7 1727  ax-9 1759  ax-10 1774  ax-11 1779  ax-12 1791  ax-13 1942  ax-ext 2414  ax-sep 4401  ax-nul 4409  ax-pr 4519

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 960  df-tru 1365  df-ex 1590  df-nf 1593  df-sb 1700  df-eu 2258  df-mo 2259  df-clab 2420  df-cleq 2426  df-clel 2429  df-nfc 2558  df-ne 2598  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2964  df-sbc 3176  df-dif 3319  df-un 3321  df-in 3323  df-ss 3330  df-nul 3626  df-if 3780  df-sn 3866  df-pr 3868  df-op 3872  df-uni 4080  df-br 4281  df-opab 4339  df-id 4623  df-xp 4833  df-rel 4834  df-cnv 4835  df-co 4836  df-dm 4837  df-rn 4838  df-res 4839  df-ima 4840  df-iota 5369  df-fun 5408  df-fn 5409  df-fv 5414

This theorem is referenced by:  fnimapr  5743  funfv  5746  fvco2  5754  fvimacnvi  5805  fvimacnvALT  5810  fsn2  5870  fparlem3  6663  fparlem4  6664  suppval1  6685  suppsnop  6693  domunsncan  7399  phplem4  7481  domunfican  7572  fiint  7576  infdifsn  7850  cantnfp1lem3  7876  cantnfp1lem3OLD  7902  symgfixelsi  15919  dprdf1o  16502  frlmlbs  18066  f1lindf  18092  cnt1  18795  xkohaus  19067  xkoptsub  19068  ustuqtop3  19659  2pthlem2  23317  eupath2lem3  23422  eulerpartlemmf  26605  grpokerinj  28591  funcoressn  29876