Theorem iunss 4007

Description: Subset theorem for an indexed union. (Contributed by NM, 13-Sep-2003.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)

Assertion

Ref	Expression
iunss	⊢ (∪x ∈ A B ⊆ C ↔ ∀x ∈ A B ⊆ C)

Distinct variable group:   x,C

Allowed substitution hints:   A(x)   B(x)

Proof of Theorem iunss

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-iun 3971	. . 3 ⊢ ∪x ∈ A B = {y ∣ ∃x ∈ A y ∈ B}
2	1	sseq1i 3295	. 2 ⊢ (∪x ∈ A B ⊆ C ↔ {y ∣ ∃x ∈ A y ∈ B} ⊆ C)
3		abss 3335	. 2 ⊢ ({y ∣ ∃x ∈ A y ∈ B} ⊆ C ↔ ∀y(∃x ∈ A y ∈ B → y ∈ C))
4		dfss2 3262	. . . 4 ⊢ (B ⊆ C ↔ ∀y(y ∈ B → y ∈ C))
5	4	ralbii 2638	. . 3 ⊢ (∀x ∈ A B ⊆ C ↔ ∀x ∈ A ∀y(y ∈ B → y ∈ C))
6		ralcom4 2877	. . 3 ⊢ (∀x ∈ A ∀y(y ∈ B → y ∈ C) ↔ ∀y∀x ∈ A (y ∈ B → y ∈ C))
7		r19.23v 2730	. . . 4 ⊢ (∀x ∈ A (y ∈ B → y ∈ C) ↔ (∃x ∈ A y ∈ B → y ∈ C))
8	7	albii 1566	. . 3 ⊢ (∀y∀x ∈ A (y ∈ B → y ∈ C) ↔ ∀y(∃x ∈ A y ∈ B → y ∈ C))
9	5, 6, 8	3bitrri 263	. 2 ⊢ (∀y(∃x ∈ A y ∈ B → y ∈ C) ↔ ∀x ∈ A B ⊆ C)
10	2, 3, 9	3bitri 262	1 ⊢ (∪x ∈ A B ⊆ C ↔ ∀x ∈ A B ⊆ C)

Syntax hints:   → wi 4   ↔ wb 176  ∀wal 1540   ∈ wcel 1710  {cab 2339  ∀wral 2614  ∃wrex 2615   ⊆ wss 3257  ∪ciun 3969

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-ral 2619  df-rex 2620  df-v 2861  df-nin 3211  df-compl 3212  df-in 3213  df-ss 3259  df-iun 3971

This theorem is referenced by:  iunss2  4011