Theorem iunxdif2 5015

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 5011	. . 3 ⊢ (∀𝑥 ∈ 𝐴 ∃𝑦 ∈ (𝐴 ∖ 𝐵)𝐶 ⊆ 𝐷 → ∪ 𝑥 ∈ 𝐴 𝐶 ⊆ ∪ 𝑦 ∈ (𝐴 ∖ 𝐵)𝐷)
2		difss 4075	. . . . 5 ⊢ (𝐴 ∖ 𝐵) ⊆ 𝐴
3		iunss1 4969	. . . . 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 5000	. . . 4 ⊢ ∪ 𝑥 ∈ 𝐴 𝐶 = ∪ 𝑦 ∈ 𝐴 𝐷
7	4, 6	sseqtrri 3962	. . 3 ⊢ ∪ 𝑦 ∈ (𝐴 ∖ 𝐵)𝐷 ⊆ ∪ 𝑥 ∈ 𝐴 𝐶
8	1, 7	jctil 529	. 2 ⊢ (∀𝑥 ∈ 𝐴 ∃𝑦 ∈ (𝐴 ∖ 𝐵)𝐶 ⊆ 𝐷 → (∪ 𝑦 ∈ (𝐴 ∖ 𝐵)𝐷 ⊆ ∪ 𝑥 ∈ 𝐴 𝐶 ∧ ∪ 𝑥 ∈ 𝐴 𝐶 ⊆ ∪ 𝑦 ∈ (𝐴 ∖ 𝐵)𝐷))
9		eqss 3928	. 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 237	1 ⊢ (∀𝑥 ∈ 𝐴 ∃𝑦 ∈ (𝐴 ∖ 𝐵)𝐶 ⊆ 𝐷 → ∪ 𝑦 ∈ (𝐴 ∖ 𝐵)𝐷 = ∪ 𝑥 ∈ 𝐴 𝐶)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570  ∀wral 3080  ∃wrex 3090   ⊆ wss 3900   ∖ cdif 4048  ∪ ciun 4954

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-11 2197  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-ral 3081  df-rex 3091  df-v 3456  df-ss 3901  df-dif 4050  df-iun 4956

This theorem is used by: (None)