Theorem imasng 3408

Description: The image of a singleton.

Assertion

Ref	Expression
imasng	|- (A e. B -> (R"{A}) = {y | ARy})

Distinct variable groups:   y,A   y,R

Proof of Theorem imasng

Step	Hyp	Ref	Expression
1		elisset 1808	. 2 |- (A e. B -> A e. V)
2		breq1 2612	. . . . . 6 |- (x = A -> (xRy <-> ARy))
3	2	ceqsexgv 1879	. . . . 5 |- (A e. V -> (E.x(x = A /\ xRy) <-> ARy))
4		df-rex 1642	. . . . . 6 |- (E.x e. {A}xRy <-> E.x(x e. {A} /\ xRy))
5		elsn 2411	. . . . . . . 8 |- (x e. {A} <-> x = A)
6	5	anbi1i 480	. . . . . . 7 |- ((x e. {A} /\ xRy) <-> (x = A /\ xRy))
7	6	exbii 1047	. . . . . 6 |- (E.x(x e. {A} /\ xRy) <-> E.x(x = A /\ xRy))
8	4, 7	bitr 173	. . . . 5 |- (E.x e. {A}xRy <-> E.x(x = A /\ xRy))
9	3, 8	syl5bb 530	. . . 4 |- (A e. V -> (E.x e. {A}xRy <-> ARy))
10	9	abbidv 1569	. . 3 |- (A e. V -> {y | E.x e. {A}xRy} = {y | ARy})
11		dfima2 3389	. . 3 |- (R"{A}) = {y | E.x e. {A}xRy}
12	10, 11	syl5eq 1511	. 2 |- (A e. V -> (R"{A}) = {y | ARy})
13	1, 12	syl 10	1 |- (A e. B -> (R"{A}) = {y | ARy})

Syntax hints:   -> wi 3   /\ wa 223   = wceq 953   e. wcel 955  E.wex 977  {cab 1456  E.wrex 1638  Vcvv 1802  {csn 2399   class class class wbr 2609  "cima 3163

This theorem is referenced by:  relimasn 3409  args 3412  aceq3 4705

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-cnv 3176  df-dm 3178  df-rn 3179  df-res 3180  df-ima 3181

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