Theorem fnsnfv 3773

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 1816	. . . . 5 |- y e. V
2	1	fnbrfvb 3759	. . . 4 |- ((F Fn A /\ B e. A) -> ((F` B) = y <-> BFy))
3		eqcom 1480	. . . 4 |- (y = (F` B) <-> (F` B) = y)
4	2, 3	syl5bb 534	. . 3 |- ((F Fn A /\ B e. A) -> (y = (F` B) <-> BFy))
5	4	abbidv 1580	. 2 |- ((F Fn A /\ B e. A) -> {y | y = (F` B)} = {y | BFy})
6		df-sn 2416	. . 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 3592	. . . 4 |- (F Fn A -> Rel F)
9		relimasn 3431	. . . 4 |- (Rel F -> (F"{B}) = {y | BFy})
10	8, 9	syl 10	. . 3 |- (F Fn A -> (F"{B}) = {y | BFy})
11	10	adantr 391	. 2 |- ((F Fn A /\ B e. A) -> (F"{B}) = {y | BFy})
12	5, 7, 11	3eqtr4d 1520	1 |- ((F Fn A /\ B e. A) -> {(F` B)} = (F"{B}))

Syntax hints:   -> wi 3   /\ wa 223   = wceq 958   e. wcel 960  {cab 1466  {csn 2413   class class class wbr 2624  "cima 3179  Rel wrel 3181   Fn wfn 3183  ` cfv 3188

This theorem is referenced by:  funfv 3776  fvimacnvi 3810  fvimacnvALT 3815  fsn2 3842  phplem4 4517  unifiOLD 4570  fiint 4572  fiintOLD 4573  fodomfi 4575  fodomfiOLD 4576

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-10 968  ax-11 969  ax-12 970  ax-13 971  ax-14 972  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462  ax-sep 2708  ax-pow 2748  ax-pr 2785

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 983  df-sb 1174  df-eu 1384  df-mo 1385  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-rex 1653  df-v 1815  df-dif 2052  df-un 2053  df-in 2054  df-ss 2056  df-nul 2284  df-pw 2406  df-sn 2416  df-pr 2417  df-op 2420  df-uni 2508  df-br 2625  df-opab 2672  df-id 2841  df-xp 3190  df-rel 3191  df-cnv 3192  df-co 3193  df-dm 3194  df-rn 3195  df-res 3196  df-ima 3197  df-fun 3198  df-fn 3199  df-fv 3204

Copyright terms: Public domain