Theorem fnsnfv 5865

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 2411	. . . 4 |- ( y = ( F ` B ) <-> ( F ` B ) = y )
2		fnbrfvb 5845	. . . 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 2538	. 2 |- ( ( F Fn A /\ B e. A ) -> { y | y = ( F ` B ) } = { y | B F y } )
5		df-sn 3972	. . 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 5616	. . . 4 |- ( F Fn A -> Rel F )
8		relimasn 5301	. . . 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 463	. 2 |- ( ( F Fn A /\ B e. A ) -> ( F " { B } ) = { y | B F y } )
11	4, 6, 10	3eqtr4d 2453	1 |- ( ( F Fn A /\ B e. A ) -> { ( F ` B ) } = ( F " { B } ) )

Syntax hints:   -> wi 4   /\ wa 367   = wceq 1405   e. wcel 1842   {cab 2387   {csn 3971   class class class wbr 4394   "cima 4945   Rel wrel 4947   Fn wfn 5520   ` cfv 5525

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-sep 4516  ax-nul 4524  ax-pr 4629

This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2758  df-rex 2759  df-rab 2762  df-v 3060  df-sbc 3277  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-nul 3738  df-if 3885  df-sn 3972  df-pr 3974  df-op 3978  df-uni 4191  df-br 4395  df-opab 4453  df-id 4737  df-xp 4948  df-rel 4949  df-cnv 4950  df-co 4951  df-dm 4952  df-rn 4953  df-res 4954  df-ima 4955  df-iota 5489  df-fun 5527  df-fn 5528  df-fv 5533

This theorem is referenced by:  fnimapr  5869  funfv  5872  fvco2  5880  fvimacnvi  5935  fvimacnvALT  5940  fsn2  6005  fparlem3  6840  fparlem4  6841  suppval1  6862  suppsnop  6870  domunsncan  7575  phplem4  7657  domunfican  7747  fiint  7751  infdifsn  8026  cantnfp1lem3  8051  cantnfp1lem3OLD  8077  symgfixelsi  16676  dprdf1o  17291  frlmlbs  19016  f1lindf  19041  cnt1  20036  xkohaus  20338  xkoptsub  20339  ustuqtop3  20930  2pthlem2  24896  eupath2lem3  25277  eulerpartlemmf  28700  grpokerinj  31610  funcoressn  37562