Theorem relimasn 3409

Description: The image of a singleton.

Assertion

Ref	Expression
relimasn	|- (Rel R -> (R"{A}) = {y | ARy})

Distinct variable groups:   y,A   y,R

Proof of Theorem relimasn

Step	Hyp	Ref	Expression
1		snprc 2433	. . . . . . 7 |- (-. A e. V <-> {A} = (/))
2		imaeq2 3386	. . . . . . 7 |- ({A} = (/) -> (R"{A}) = (R"(/)))
3	1, 2	sylbi 199	. . . . . 6 |- (-. A e. V -> (R"{A}) = (R"(/)))
4		ima0 3404	. . . . . 6 |- (R"(/)) = (/)
5	3, 4	syl6eq 1515	. . . . 5 |- (-. A e. V -> (R"{A}) = (/))
6	5	adantl 388	. . . 4 |- ((Rel R /\ -. A e. V) -> (R"{A}) = (/))
7		brrelex 3197	. . . . . . . . 9 |- ((Rel R /\ ARy) -> A e. V)
8	7	ex 373	. . . . . . . 8 |- (Rel R -> (ARy -> A e. V))
9	8	con3d 95	. . . . . . 7 |- (Rel R -> (-. A e. V -> -. ARy))
10	9	imp 350	. . . . . 6 |- ((Rel R /\ -. A e. V) -> -. ARy)
11	10	nexdv 1321	. . . . 5 |- ((Rel R /\ -. A e. V) -> -. E.y ARy)
12		abn0 2280	. . . . . 6 |- ({y | ARy} =/= (/) <-> E.y ARy)
13	12	necon1bbii 1609	. . . . 5 |- (-. E.y ARy <-> {y | ARy} = (/))
14	11, 13	sylib 198	. . . 4 |- ((Rel R /\ -. A e. V) -> {y | ARy} = (/))
15	6, 14	eqtr4d 1502	. . 3 |- ((Rel R /\ -. A e. V) -> (R"{A}) = {y | ARy})
16	15	ex 373	. 2 |- (Rel R -> (-. A e. V -> (R"{A}) = {y | ARy}))
17		imasng 3408	. 2 |- (A e. V -> (R"{A}) = {y | ARy})
18	16, 17	pm2.61d2 129	1 |- (Rel R -> (R"{A}) = {y | ARy})

Syntax hints:  -. wn 2   -> wi 3   /\ wa 223   = wceq 953   e. wcel 955  E.wex 977  {cab 1456  Vcvv 1802  (/)c0 2270  {csn 2399   class class class wbr 2609  "cima 3163  Rel wrel 3165

This theorem is referenced by:  fnsnfv 3752  funfv2 3756  mapsn 4329

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-10 963  ax-11 964  ax-12 965  ax-13 966  ax-14 967  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452  ax-sep 2693  ax-pow 2732  ax-pr 2769

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 978  df-sb 1168  df-eu 1375  df-mo 1376  df-clab 1457  df-cleq 1462  df-clel 1465  df-ne 1579  df-rex 1642  df-v 1803  df-dif 2039  df-un 2040  df-in 2041  df-ss 2043  df-nul 2271  df-pw 2392  df-sn 2402  df-pr 2403  df-op 2406  df-br 2610  df-opab 2657  df-xp 3174  df-rel 3175  df-cnv 3176  df-dm 3178  df-rn 3179  df-res 3180  df-ima 3181

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