Theorem iunxsn 2602

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 2560	. . 3 |- (y e. U_x e. {A}B <-> E.x e. {A}y e. B)
2		df-rex 1642	. . 3 |- (E.x e. {A}y e. B <-> E.x(x e. {A} /\ y e. B))
3		elsn 2411	. . . . . 6 |- (x e. {A} <-> x = A)
4	3	anbi1i 480	. . . . 5 |- ((x e. {A} /\ y e. B) <-> (x = A /\ y e. B))
5	4	exbii 1047	. . . 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 1533	. . . . 5 |- (x = A -> (y e. B <-> y e. C))
9	6, 8	ceqsexv 1826	. . . 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 1467	1 |- U_x e. {A}B = C

Syntax hints:   -> wi 3   /\ wa 223   = wceq 953   e. wcel 955  E.wex 977  E.wrex 1638  Vcvv 1802  {csn 2399  U_ciun 2556

This theorem is referenced by:  kmlem11 4747

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-12 965  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

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 978  df-sb 1168  df-clab 1457  df-cleq 1462  df-clel 1465  df-rex 1642  df-v 1803  df-sn 2402  df-iun 2558

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