Theorem fnsnfv 5592

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 2191	. . . 4 |- ( y = ( F ` B ) <-> ( F ` B ) = y )
2		fnbrfvb 5573	. . . 4 |- ( ( F Fn A /\ B e. A ) -> ( ( F ` B ) = y <-> B F y ) )
3	1, 2	bitrid 192	. . 3 |- ( ( F Fn A /\ B e. A ) -> ( y = ( F ` B ) <-> B F y ) )
4	3	abbidv 2307	. 2 |- ( ( F Fn A /\ B e. A ) -> { y | y = ( F ` B ) } = { y | B F y } )
5		df-sn 3613	. . 3 |- { ( F ` B ) } = { y | y = ( F ` B ) }
6	5	a1i 9	. 2 |- ( ( F Fn A /\ B e. A ) -> { ( F ` B ) } = { y | y = ( F ` B ) } )
7		imasng 5008	. . 3 |- ( B e. A -> ( F " { B } ) = { y | B F y } )
8	7	adantl 277	. 2 |- ( ( F Fn A /\ B e. A ) -> ( F " { B } ) = { y | B F y } )
9	4, 6, 8	3eqtr4d 2232	1 |- ( ( F Fn A /\ B e. A ) -> { ( F ` B ) } = ( F " { B } ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1364   e. wcel 2160   {cab 2175   {csn 3607   class class class wbr 4018   "cima 4644   Fn wfn 5227   ` cfv 5232

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4189  ax-pr 4224

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-v 2754  df-sbc 2978  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-br 4019  df-opab 4080  df-id 4308  df-xp 4647  df-rel 4648  df-cnv 4649  df-co 4650  df-dm 4651  df-rn 4652  df-res 4653  df-ima 4654  df-iota 5193  df-fun 5234  df-fn 5235  df-fv 5240

This theorem is referenced by:  fnimapr  5593  funfvdm  5596  fvco2  5602  fvimacnvi  5647  fsn2  5707  phplem4  6878  phplem4dom  6885  phplem4on  6890  fiintim  6952  fidcenumlemrks  6977  fidcenumlemr  6979  resunimafz0  10838  ennnfonelemhf1o  12459