Theorem iunxunpr 30808

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 5019	. 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 5021	. . 3 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊) → ∪ 𝑥 ∈ {𝑋, 𝑌}𝐵 = (𝐶 ∪ 𝐷))
5	4	uneq2d 4093	. 2 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊) → (∪ 𝑥 ∈ 𝐴 𝐵 ∪ ∪ 𝑥 ∈ {𝑋, 𝑌}𝐵) = (∪ 𝑥 ∈ 𝐴 𝐵 ∪ (𝐶 ∪ 𝐷)))
6	1, 5	syl5eq 2791	1 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊) → ∪ 𝑥 ∈ (𝐴 ∪ {𝑋, 𝑌})𝐵 = (∪ 𝑥 ∈ 𝐴 𝐵 ∪ (𝐶 ∪ 𝐷)))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1539   ∈ wcel 2108   ∪ cun 3881  {cpr 4560  ∪ ciun 4921

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-tru 1542  df-ex 1784  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817  df-ral 3068  df-rex 3069  df-v 3424  df-un 3888  df-in 3890  df-ss 3900  df-sn 4559  df-pr 4561  df-iun 4923

This theorem is referenced by: (None)