Theorem relimasn 3425

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 2443	. . . . . . 7 |- (-. A e. V <-> {A} = (/))
2		imaeq2 3402	. . . . . . 7 |- ({A} = (/) -> (R"{A}) = (R"(/)))
3	1, 2	sylbi 199	. . . . . 6 |- (-. A e. V -> (R"{A}) = (R"(/)))
4		ima0 3420	. . . . . 6 |- (R"(/)) = (/)
5	3, 4	syl6eq 1523	. . . . 5 |- (-. A e. V -> (R"{A}) = (/))
6	5	adantl 388	. . . 4 |- ((Rel R /\ -. A e. V) -> (R"{A}) = (/))
7		brrelex 3207	. . . . . . . . 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 1326	. . . . 5 |- ((Rel R /\ -. A e. V) -> -. E.y ARy)
12		abn0 2290	. . . . . 6 |- ({y | ARy} =/= (/) <-> E.y ARy)
13	12	necon1bbii 1617	. . . . 5 |- (-. E.y ARy <-> {y | ARy} = (/))
14	11, 13	sylib 198	. . . 4 |- ((Rel R /\ -. A e. V) -> {y | ARy} = (/))
15	6, 14	eqtr4d 1510	. . 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 3424	. 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 956   e. wcel 958  E.wex 980  {cab 1463  Vcvv 1811  (/)c0 2280  {csn 2409   class class class wbr 2619  "cima 3173  Rel wrel 3175

This theorem is referenced by:  fnsnfv 3767  funfv2 3771  mapsn 4345

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-br 2620  df-opab 2667  df-xp 3184  df-rel 3185  df-cnv 3186  df-dm 3188  df-rn 3189  df-res 3190  df-ima 3191

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