Theorem iunxdif2 4014

Description: Indexed union with a class difference as its index. (Contributed by NM, 10-Dec-2004.)

Hypothesis

Ref	Expression
iunxdif2.1	⊢ (x = y → C = D)

Assertion

Ref	Expression
iunxdif2	⊢ (∀x ∈ A ∃y ∈ (A ∖ B)C ⊆ D → ∪y ∈ (A ∖ B)D = ∪x ∈ A C)

Distinct variable groups:   x,y,A   x,B,y   y,C   x,D

Allowed substitution hints:   C(x)   D(y)

Proof of Theorem iunxdif2

Step	Hyp	Ref	Expression
1		iunss2 4011	. . 3 ⊢ (∀x ∈ A ∃y ∈ (A ∖ B)C ⊆ D → ∪x ∈ A C ⊆ ∪y ∈ (A ∖ B)D)
2		difss 3393	. . . . 5 ⊢ (A ∖ B) ⊆ A
3		iunss1 3980	. . . . 5 ⊢ ((A ∖ B) ⊆ A → ∪y ∈ (A ∖ B)D ⊆ ∪y ∈ A D)
4	2, 3	ax-mp 8	. . . 4 ⊢ ∪y ∈ (A ∖ B)D ⊆ ∪y ∈ A D
5		iunxdif2.1	. . . . 5 ⊢ (x = y → C = D)
6	5	cbviunv 4005	. . . 4 ⊢ ∪x ∈ A C = ∪y ∈ A D
7	4, 6	sseqtr4i 3304	. . 3 ⊢ ∪y ∈ (A ∖ B)D ⊆ ∪x ∈ A C
8	1, 7	jctil 523	. 2 ⊢ (∀x ∈ A ∃y ∈ (A ∖ B)C ⊆ D → (∪y ∈ (A ∖ B)D ⊆ ∪x ∈ A C ∧ ∪x ∈ A C ⊆ ∪y ∈ (A ∖ B)D))
9		eqss 3287	. 2 ⊢ (∪y ∈ (A ∖ B)D = ∪x ∈ A C ↔ (∪y ∈ (A ∖ B)D ⊆ ∪x ∈ A C ∧ ∪x ∈ A C ⊆ ∪y ∈ (A ∖ B)D))
10	8, 9	sylibr 203	1 ⊢ (∀x ∈ A ∃y ∈ (A ∖ B)C ⊆ D → ∪y ∈ (A ∖ B)D = ∪x ∈ A C)

Syntax hints:   → wi 4   ∧ wa 358   = wceq 1642  ∀wral 2614  ∃wrex 2615   ∖ cdif 3206   ⊆ 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-dif 3215  df-ss 3259  df-iun 3971

This theorem is referenced by: (None)