Theorem iunxunpr 33230

Description: Appending two sets to an indexed union. (Contributed by Thierry Arnoux, 20-Nov-2023.)

Hypotheses

Ref	Expression
iunxunsn.1	⊢ (𝑥 = 𝑋 → 𝐵 = 𝐶)
iunxunpr.2	⊢ (𝑥 = 𝑌 → 𝐵 = 𝐷)

Assertion

Ref	Expression
iunxunpr	⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊) → ∪ 𝑥 ∈ (𝐴 ∪ {𝑋, 𝑌})𝐵 = (∪ 𝑥 ∈ 𝐴 𝐵 ∪ (𝐶 ∪ 𝐷)))

Distinct variable groups:   𝑥,𝐶   𝑥,𝑋   𝑥,𝐷   𝑥,𝑌

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

Proof of Theorem iunxunpr

Step	Hyp	Ref	Expression
1		iunxun 5057	. 2 ⊢ ∪ 𝑥 ∈ (𝐴 ∪ {𝑋, 𝑌})𝐵 = (∪ 𝑥 ∈ 𝐴 𝐵 ∪ ∪ 𝑥 ∈ {𝑋, 𝑌}𝐵)
2		iunxunsn.1	. . . 4 ⊢ (𝑥 = 𝑋 → 𝐵 = 𝐶)
3		iunxunpr.2	. . . 4 ⊢ (𝑥 = 𝑌 → 𝐵 = 𝐷)
4	2, 3	iunxprg 5059	. . 3 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊) → ∪ 𝑥 ∈ {𝑋, 𝑌}𝐵 = (𝐶 ∪ 𝐷))
5	4	uneq2d 4170	. 2 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊) → (∪ 𝑥 ∈ 𝐴 𝐵 ∪ ∪ 𝑥 ∈ {𝑋, 𝑌}𝐵) = (∪ 𝑥 ∈ 𝐴 𝐵 ∪ (𝐶 ∪ 𝐷)))
6	1, 5	eqtrid 2810	1 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊) → ∪ 𝑥 ∈ (𝐴 ∪ {𝑋, 𝑌})𝐵 = (∪ 𝑥 ∈ 𝐴 𝐵 ∪ (𝐶 ∪ 𝐷)))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2147   ∪ cun 4152  {cpr 4590  ∪ 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-ext 2735

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  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-un 4154  df-sn 4589  df-pr 4591  df-iun 4956

This theorem is used by: (None)