Theorem iunxsn 2034

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

Hypotheses

Ref	Expression
iunxsn.1	⊢ A ∈ V
iunxsn.2	⊢ (x = A → B = C)

Assertion

Ref	Expression
iunxsn	⊢ ∪x ∈ {A}B = C

Distinct variable group(s):   x,A   x,C

Proof of Theorem iunxsn

Step	Hyp	Ref	Expression
1		eliun 1998	. . 3 ⊢ (y ∈ ∪x ∈ {A}B ↔ ∃x ∈ {A}y ∈ B)
2		df-rex 1206	. . . 4 ⊢ (∃x ∈ {A}y ∈ B ↔ ∃x(x ∈ {A} ∧ y ∈ B))
3		elsn 1820	. . . . . . 7 ⊢ (x ∈ {A} ↔ x = A)
4	3	anbi1i 368	. . . . . 6 ⊢ ((x ∈ {A} ∧ y ∈ B) ↔ (x = A ∧ y ∈ B))
5	4	biex 733	. . . . 5 ⊢ (∃x(x ∈ {A} ∧ y ∈ B) ↔ ∃x(x = A ∧ y ∈ B))
6		iunxsn.1	. . . . . 6 ⊢ A ∈ V
7		iunxsn.2	. . . . . . 7 ⊢ (x = A → B = C)
8	7	eleq2d 1156	. . . . . 6 ⊢ (x = A → (y ∈ B ↔ y ∈ C))
9	6, 8	ceqsexv 1371	. . . . 5 ⊢ (∃x(x = A ∧ y ∈ B) ↔ y ∈ C)
10	5, 9	bitr 151	. . . 4 ⊢ (∃x(x ∈ {A} ∧ y ∈ B) ↔ y ∈ C)
11	2, 10	bitr 151	. . 3 ⊢ (∃x ∈ {A}y ∈ B ↔ y ∈ C)
12	1, 11	bitr 151	. 2 ⊢ (y ∈ ∪x ∈ {A}B ↔ y ∈ C)
13	12	cleqri 1101	1 ⊢ ∪x ∈ {A}B = C

Syntax hints:   → wi 2   ∧ wa 196  ∃wex 678   = wceq 1091   ∈ wcel 1092  ∃wrex 1202  Vcvv 1348  {csn 1808  ∪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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