Theorem fnsnfv 2861

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 1350	. . . . 5 |- y e. V
2	1	fnfvop 2856	. . . 4 |- ((F Fn A /\ B e. A) -> ((F` B) = y <-> <.B, y>. e. F))
3		cleqcom 1103	. . . 4 |- (y = (F` B) <-> (F` B) = y)
4	2, 3	syl5bb 410	. . 3 |- ((F Fn A /\ B e. A) -> (y = (F` B) <-> <.B, y>. e. F))
5	4	biabdv 1183	. 2 |- ((F Fn A /\ B e. A) -> {y | y = (F` B)} = {y | <.B, y>. e. F})
6		df-sn 1811	. . 3 |- {(F` B)} = {y | y = (F` B)}
7	6	a1i 7	. 2 |- ((F Fn A /\ B e. A) -> {(F` B)} = {y | y = (F` B)})
8		fnrel 2722	. . . 4 |- (F Fn A -> Rel F)
9		imasn 2616	. . . 4 |- (Rel F -> (F"{B}) = {y | <.B, y>. e. F})
10	8, 9	syl 12	. . 3 |- (F Fn A -> (F"{B}) = {y | <.B, y>. e. F})
11	10	adantr 306	. 2 |- ((F Fn A /\ B e. A) -> (F"{B}) = {y | <.B, y>. e. F})
12	5, 7, 11	3eqtr4d 1134	1 |- ((F Fn A /\ B e. A) -> {(F` B)} = (F"{B}))

Syntax hints:   -> wi 2   /\ wa 196  {cab 1090   = wceq 1091   e. wcel 1092  {csn 1808  <.cop 1810  "cima 2413  Rel wrel 2415   Fn wfn 2417  ` cfv 2422

This theorem is referenced by:  funfv 2862  fsn2 2896  phplem5 3407  fiint 3445

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-eu 1009  df-mo 1010  df-clab 1093  df-cleq 1097  df-clel 1099  df-rex 1206  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-op 1815  df-uni 1920  df-br 2063  df-opab 2098  df-id 2125  df-xp 2424  df-rel 2425  df-cnv 2426  df-co 2427  df-dm 2428  df-rn 2429  df-res 2430  df-ima 2431  df-fun 2432  df-fn 2433  df-fv 2438

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