Theorem fnsnfv 4728

Description: Singleton of function value.

Assertion

Ref	Expression
fnsnfv	|- ((F Fn A /\ B e. A) -> {(F` B)} = (F"{B}))

Proof of Theorem fnsnfv

Step	Hyp	Ref	Expression
1		visset 2295	. . . . 5 |- y e. _V
2	1	fnbrfvb 4712	. . . 4 |- ((F Fn A /\ B e. A) -> ((F` B) = y <-> BFy))
3		eqcom 1886	. . . 4 |- (y = (F` B) <-> (F` B) = y)
4	2, 3	syl5bb 591	. . 3 |- ((F Fn A /\ B e. A) -> (y = (F` B) <-> BFy))
5	4	abbidv 2008	. 2 |- ((F Fn A /\ B e. A) -> {y | y = (F` B)} = {y | BFy})
6		df-sn 3049	. . 3 |- {(F` B)} = {y | y = (F` B)}
7	6	a1i 8	. 2 |- ((F Fn A /\ B e. A) -> {(F` B)} = {y | y = (F` B)})
8		fnrel 4511	. . . 4 |- (F Fn A -> Rel F)
9		relimasn 4288	. . . 4 |- (Rel F -> (F"{B}) = {y | BFy})
10	8, 9	syl 12	. . 3 |- (F Fn A -> (F"{B}) = {y | BFy})
11	10	adantr 425	. 2 |- ((F Fn A /\ B e. A) -> (F"{B}) = {y | BFy})
12	5, 7, 11	3eqtr4d 1937	1 |- ((F Fn A /\ B e. A) -> {(F` B)} = (F"{B}))

Syntax hints:   -> wi 3   /\ wa 240   = wceq 1298   e. wcel 1300  {cab 1871  {csn 3044   class class class wbr 3338  "cima 3989  Rel wrel 3991   Fn wfn 3993  ` cfv 3998

This theorem is referenced by:  funfv 4731  fvimacnvi 4777  fvimacnvALT 4782  fsn2 4809  fparlem3 5083  fparlem4 5084  ac6sfilem2 5507  ac6sfi 5509  phplem4 5605  unifi 5648  fiint 5650  fodomfi 5656  fbssint 10279  finsschain 15373  neibastop2lem4 15522  fcluscomplem 15620  fnimapr 15687  grpkerinj 16042

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-rex 2110  df-v 2294  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-nul 2876  df-pw 3035  df-sn 3049  df-pr 3050  df-op 3053  df-uni 3178  df-br 3339  df-opab 3396  df-id 3586  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fn 4009  df-fv 4014

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