Theorem iunxdif2 5059

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 5055	. . 3 ⊢ (∀𝑥 ∈ 𝐴 ∃𝑦 ∈ (𝐴 ∖ 𝐵)𝐶 ⊆ 𝐷 → ∪ 𝑥 ∈ 𝐴 𝐶 ⊆ ∪ 𝑦 ∈ (𝐴 ∖ 𝐵)𝐷)
2		difss 4147	. . . . 5 ⊢ (𝐴 ∖ 𝐵) ⊆ 𝐴
3		iunss1 5012	. . . . 5 ⊢ ((𝐴 ∖ 𝐵) ⊆ 𝐴 → ∪ 𝑦 ∈ (𝐴 ∖ 𝐵)𝐷 ⊆ ∪ 𝑦 ∈ 𝐴 𝐷)
4	2, 3	ax-mp 5	. . . 4 ⊢ ∪ 𝑦 ∈ (𝐴 ∖ 𝐵)𝐷 ⊆ ∪ 𝑦 ∈ 𝐴 𝐷
5		iunxdif2.1	. . . . 5 ⊢ (𝑥 = 𝑦 → 𝐶 = 𝐷)
6	5	cbviunv 5046	. . . 4 ⊢ ∪ 𝑥 ∈ 𝐴 𝐶 = ∪ 𝑦 ∈ 𝐴 𝐷
7	4, 6	sseqtrri 4034	. . 3 ⊢ ∪ 𝑦 ∈ (𝐴 ∖ 𝐵)𝐷 ⊆ ∪ 𝑥 ∈ 𝐴 𝐶
8	1, 7	jctil 519	. 2 ⊢ (∀𝑥 ∈ 𝐴 ∃𝑦 ∈ (𝐴 ∖ 𝐵)𝐶 ⊆ 𝐷 → (∪ 𝑦 ∈ (𝐴 ∖ 𝐵)𝐷 ⊆ ∪ 𝑥 ∈ 𝐴 𝐶 ∧ ∪ 𝑥 ∈ 𝐴 𝐶 ⊆ ∪ 𝑦 ∈ (𝐴 ∖ 𝐵)𝐷))
9		eqss 4012	. 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 234	1 ⊢ (∀𝑥 ∈ 𝐴 ∃𝑦 ∈ (𝐴 ∖ 𝐵)𝐶 ⊆ 𝐷 → ∪ 𝑦 ∈ (𝐴 ∖ 𝐵)𝐷 = ∪ 𝑥 ∈ 𝐴 𝐶)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1538  ∀wral 3060  ∃wrex 3069   ∖ cdif 3961   ⊆ wss 3964  ∪ ciun 4997

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1541  df-ex 1778  df-nf 1782  df-sb 2064  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ral 3061  df-rex 3070  df-v 3481  df-dif 3967  df-ss 3981  df-iun 4999

This theorem is referenced by: (None)