Theorem imasng 3424

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 1817	. 2 |- (A e. B -> A e. V)
2		breq1 2622	. . . . . 6 |- (x = A -> (xRy <-> ARy))
3	2	ceqsexgv 1888	. . . . 5 |- (A e. V -> (E.x(x = A /\ xRy) <-> ARy))
4		df-rex 1650	. . . . . 6 |- (E.x e. {A}xRy <-> E.x(x e. {A} /\ xRy))
5		elsn 2421	. . . . . . . 8 |- (x e. {A} <-> x = A)
6	5	anbi1i 481	. . . . . . 7 |- ((x e. {A} /\ xRy) <-> (x = A /\ xRy))
7	6	exbii 1051	. . . . . 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 532	. . . 4 |- (A e. V -> (E.x e. {A}xRy <-> ARy))
10	9	abbidv 1577	. . 3 |- (A e. V -> {y | E.x e. {A}xRy} = {y | ARy})
11		dfima2 3405	. . 3 |- (R"{A}) = {y | E.x e. {A}xRy}
12	10, 11	syl5eq 1519	. 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 956   e. wcel 958  E.wex 980  {cab 1463  E.wrex 1646  Vcvv 1811  {csn 2409   class class class wbr 2619  "cima 3173

This theorem is referenced by:  relimasn 3425  args 3428  aceq3 4733

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-cnv 3186  df-dm 3188  df-rn 3189  df-res 3190  df-ima 3191

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