Theorem iunxsn 2034

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 group(s):   x,A   x,C

Proof of Theorem iunxsn

Step	Hyp	Ref	Expression
1		eliun 1998	. . 3 |- (y e. U.x e. {A}B <-> E.x e. {A}y e. B)
2		df-rex 1206	. . . 4 |- (E.x e. {A}y e. B <-> E.x(x e. {A} /\ y e. B))
3		elsn 1820	. . . . . . 7 |- (x e. {A} <-> x = A)
4	3	anbi1i 368	. . . . . 6 |- ((x e. {A} /\ y e. B) <-> (x = A /\ y e. B))
5	4	biex 733	. . . . 5 |- (E.x(x e. {A} /\ y e. B) <-> E.x(x = A /\ y e. B))
6		iunxsn.1	. . . . . 6 |- A e. V
7		iunxsn.2	. . . . . . 7 |- (x = A -> B = C)
8	7	eleq2d 1156	. . . . . 6 |- (x = A -> (y e. B <-> y e. C))
9	6, 8	ceqsexv 1371	. . . . 5 |- (E.x(x = A /\ y e. B) <-> y e. C)
10	5, 9	bitr 151	. . . 4 |- (E.x(x e. {A} /\ y e. B) <-> y e. C)
11	2, 10	bitr 151	. . 3 |- (E.x e. {A}y e. B <-> y e. C)
12	1, 11	bitr 151	. 2 |- (y e. U.x e. {A}B <-> y e. C)
13	12	cleqri 1101	1 |- U.x e. {A}B = C

Syntax hints:   -> wi 2   /\ wa 196  E.wex 678   = wceq 1091   e. wcel 1092  E.wrex 1202  Vcvv 1348  {csn 1808  U.ciun 1994

This theorem is referenced by:  kmlem10 3589

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-16 922  ax-17 925  ax-ext 1074

This theorem depends on definitions:  df-bi 128  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-rex 1206  df-v 1349  df-sn 1811  df-iun 1996

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