Theorem relimasn 4288

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 3092	. . . . . . 7 |- (-. A e. _V <-> {A} = (/))
2		imaeq2 4260	. . . . . . 7 |- ({A} = (/) -> (R"{A}) = (R"(/)))
3	1, 2	sylbi 216	. . . . . 6 |- (-. A e. _V -> (R"{A}) = (R"(/)))
4		ima0 4283	. . . . . 6 |- (R"(/)) = (/)
5	3, 4	syl6eq 1944	. . . . 5 |- (-. A e. _V -> (R"{A}) = (/))
6	5	adantl 424	. . . 4 |- ((Rel R /\ -. A e. _V) -> (R"{A}) = (/))
7		brrelex 4028	. . . . . . . . 9 |- ((Rel R /\ ARy) -> A e. _V)
8	7	ex 402	. . . . . . . 8 |- (Rel R -> (ARy -> A e. _V))
9	8	con3d 111	. . . . . . 7 |- (Rel R -> (-. A e. _V -> -. ARy))
10	9	imp 377	. . . . . 6 |- ((Rel R /\ -. A e. _V) -> -. ARy)
11	10	nexdv 1711	. . . . 5 |- ((Rel R /\ -. A e. _V) -> -. E.y ARy)
12		abn0 2892	. . . . . 6 |- ({y | ARy} =/= (/) <-> E.y ARy)
13	12	necon1bbii 2060	. . . . 5 |- (-. E.y ARy <-> {y | ARy} = (/))
14	11, 13	sylib 215	. . . 4 |- ((Rel R /\ -. A e. _V) -> {y | ARy} = (/))
15	6, 14	eqtr4d 1928	. . 3 |- ((Rel R /\ -. A e. _V) -> (R"{A}) = {y | ARy})
16	15	ex 402	. 2 |- (Rel R -> (-. A e. _V -> (R"{A}) = {y | ARy}))
17		imasng 4287	. 2 |- (A e. _V -> (R"{A}) = {y | ARy})
18	16, 17	pm2.61d2 143	1 |- (Rel R -> (R"{A}) = {y | ARy})

Syntax hints:  -. wn 2   -> wi 3   /\ wa 240   = wceq 1298   e. wcel 1300  E.wex 1326  {cab 1871  _Vcvv 2292  (/)c0 2875  {csn 3044   class class class wbr 3338  "cima 3989  Rel wrel 3991

This theorem is referenced by:  fnsnfv 4728  funfv2 4732  mapsn 5404  predep 13903

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-rex 2110  df-v 2294  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-nul 2876  df-pw 3035  df-sn 3049  df-pr 3050  df-op 3053  df-br 3339  df-opab 3396  df-xp 4000  df-rel 4001  df-cnv 4002  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007

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