Theorem iuneq1 2003

Description: Equality theorem for indexed union.

Assertion

Ref	Expression
iuneq1	⊢ (A = B → ∪x ∈ A C = ∪x ∈ B C)

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

Proof of Theorem iuneq1

Step	Hyp	Ref	Expression
1		iunss1 2002	. . 3 ⊢ (A ⊆ B → ∪x ∈ A C ⊆ ∪x ∈ B C)
2		iunss1 2002	. . 3 ⊢ (B ⊆ A → ∪x ∈ B C ⊆ ∪x ∈ A C)
3	1, 2	anim12i 268	. 2 ⊢ ((A ⊆ B ∧ B ⊆ A) → (∪x ∈ A C ⊆ ∪x ∈ B C ∧ ∪x ∈ B C ⊆ ∪x ∈ A C))
4		eqss 1516	. 2 ⊢ (A = B ↔ (A ⊆ B ∧ B ⊆ A))
5		eqss 1516	. 2 ⊢ (∪x ∈ A C = ∪x ∈ B C ↔ (∪x ∈ A C ⊆ ∪x ∈ B C ∧ ∪x ∈ B C ⊆ ∪x ∈ A C))
6	3, 4, 5	3imtr4 192	1 ⊢ (A = B → ∪x ∈ A C = ∪x ∈ B C)

Syntax hints:   → wi 2   ∧ wa 196   = wceq 1091   ⊆ wss 1487  ∪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-in 1491  df-ss 1492  df-iun 1996

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