Theorem iunxsn 2612

Description: A singleton index picks out an instance of an indexed union's argument.

Hypotheses

Ref	Expression
iunxsn.1	|- A e. V
iunxsn.2	|- (x = A -> B = C)

Assertion

Ref	Expression
iunxsn	|- U_x e. {A}B = C

Distinct variable groups:   x,A   x,C

Proof of Theorem iunxsn

Step	Hyp	Ref	Expression
1		eliun 2570	. . 3 |- (y e. U_x e. {A}B <-> E.x e. {A}y e. B)
2		df-rex 1650	. . 3 |- (E.x e. {A}y e. B <-> E.x(x e. {A} /\ y e. B))
3		elsn 2421	. . . . . 6 |- (x e. {A} <-> x = A)
4	3	anbi1i 481	. . . . 5 |- ((x e. {A} /\ y e. B) <-> (x = A /\ y e. B))
5	4	exbii 1051	. . . 4 |- (E.x(x e. {A} /\ y e. B) <-> E.x(x = A /\ y e. B))
6		iunxsn.1	. . . . 5 |- A e. V
7		iunxsn.2	. . . . . 6 |- (x = A -> B = C)
8	7	eleq2d 1541	. . . . 5 |- (x = A -> (y e. B <-> y e. C))
9	6, 8	ceqsexv 1835	. . . 4 |- (E.x(x = A /\ y e. B) <-> y e. C)
10	5, 9	bitr 173	. . 3 |- (E.x(x e. {A} /\ y e. B) <-> y e. C)
11	1, 2, 10	3bitr 177	. 2 |- (y e. U_x e. {A}B <-> y e. C)
12	11	eqriv 1474	1 |- U_x e. {A}B = C

Syntax hints:   -> wi 3   /\ wa 223   = wceq 956   e. wcel 958  E.wex 980  E.wrex 1646  Vcvv 1811  {csn 2409  U_ciun 2566

This theorem is referenced by:  kmlem11 4775

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-12 968  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

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 981  df-sb 1172  df-clab 1464  df-cleq 1469  df-clel 1472  df-rex 1650  df-v 1812  df-sn 2412  df-iun 2568

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