Theorem fnsnfv 3767

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 1813	. . . . 5 |- y e. V
2	1	fnbrfvb 3753	. . . 4 |- ((F Fn A /\ B e. A) -> ((F` B) = y <-> BFy))
3		eqcom 1477	. . . 4 |- (y = (F` B) <-> (F` B) = y)
4	2, 3	syl5bb 532	. . 3 |- ((F Fn A /\ B e. A) -> (y = (F` B) <-> BFy))
5	4	abbidv 1577	. 2 |- ((F Fn A /\ B e. A) -> {y | y = (F` B)} = {y | BFy})
6		df-sn 2412	. . 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 3586	. . . 4 |- (F Fn A -> Rel F)
9		relimasn 3425	. . . 4 |- (Rel F -> (F"{B}) = {y | BFy})
10	8, 9	syl 10	. . 3 |- (F Fn A -> (F"{B}) = {y | BFy})
11	10	adantr 389	. 2 |- ((F Fn A /\ B e. A) -> (F"{B}) = {y | BFy})
12	5, 7, 11	3eqtr4d 1517	1 |- ((F Fn A /\ B e. A) -> {(F` B)} = (F"{B}))

Syntax hints:   -> wi 3   /\ wa 223   = wceq 956   e. wcel 958  {cab 1463  {csn 2409   class class class wbr 2619  "cima 3173  Rel wrel 3175   Fn wfn 3177  ` cfv 3182

This theorem is referenced by:  funfv 3770  fvimacnvi 3804  fvimacnvALT 3809  fsn2 3836  phplem4 4511  unifiOLD 4557  fiint 4559  fiintOLD 4560  fodomfiOLD 4566

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-sep 2703  ax-pow 2742  ax-pr 2779

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-rex 1650  df-v 1812  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-nul 2281  df-pw 2402  df-sn 2412  df-pr 2413  df-op 2416  df-uni 2504  df-br 2620  df-opab 2667  df-id 2835  df-xp 3184  df-rel 3185  df-cnv 3186  df-co 3187  df-dm 3188  df-rn 3189  df-res 3190  df-ima 3191  df-fun 3192  df-fn 3193  df-fv 3198

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