Theorem iunss1 2002

Description: Subclass theorem for indexed union.

Assertion

Ref	Expression
iunss1	⊢ (A ⊆ B → ∪x ∈ A C ⊆ ∪x ∈ B C)

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

Proof of Theorem iunss1

Step	Hyp	Ref	Expression
1		ssel 1502	. . . . . 6 ⊢ (A ⊆ B → (x ∈ A → x ∈ B))
2	1	anim1d 432	. . . . 5 ⊢ (A ⊆ B → ((x ∈ A ∧ y ∈ C) → (x ∈ B ∧ y ∈ C)))
3	2	r19.22dv2 1277	. . . 4 ⊢ (A ⊆ B → (∃x ∈ A y ∈ C → ∃x ∈ B y ∈ C))
4	3	19.21aiv 943	. . 3 ⊢ (A ⊆ B → ∀y(∃x ∈ A y ∈ C → ∃x ∈ B y ∈ C))
5		ss2ab 1551	. . 3 ⊢ ({y∣∃x ∈ A y ∈ C} ⊆ {y∣∃x ∈ B y ∈ C} ↔ ∀y(∃x ∈ A y ∈ C → ∃x ∈ B y ∈ C))
6	4, 5	sylibr 175	. 2 ⊢ (A ⊆ B → {y∣∃x ∈ A y ∈ C} ⊆ {y∣∃x ∈ B y ∈ C})
7		df-iun 1996	. 2 ⊢ ∪x ∈ A C = {y∣∃x ∈ A y ∈ C}
8		df-iun 1996	. 2 ⊢ ∪x ∈ B C = {y∣∃x ∈ B y ∈ C}
9	6, 7, 8	3sstr4g 1541	1 ⊢ (A ⊆ B → ∪x ∈ A C ⊆ ∪x ∈ B C)

Syntax hints:   → wi 2  ∀wal 672  {cab 1090   ∈ wcel 1092  ∃wrex 1202   ⊆ wss 1487  ∪ciun 1994

This theorem is referenced by:  iuneq1 2003

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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