Theorem iunxunpr 30317

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 5009	. 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 5011	. . 3 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊) → ∪ 𝑥 ∈ {𝑋, 𝑌}𝐵 = (𝐶 ∪ 𝐷))
5	4	uneq2d 4132	. 2 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊) → (∪ 𝑥 ∈ 𝐴 𝐵 ∪ ∪ 𝑥 ∈ {𝑋, 𝑌}𝐵) = (∪ 𝑥 ∈ 𝐴 𝐵 ∪ (𝐶 ∪ 𝐷)))
6	1, 5	syl5eq 2867	1 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊) → ∪ 𝑥 ∈ (𝐴 ∪ {𝑋, 𝑌})𝐵 = (∪ 𝑥 ∈ 𝐴 𝐵 ∪ (𝐶 ∪ 𝐷)))

Syntax hints:   → wi 4   ∧ wa 398   = wceq 1536   ∈ wcel 2113   ∪ cun 3927  {cpr 4562  ∪ ciun 4912

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1969  ax-7 2014  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2160  ax-12 2176  ax-ext 2792

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1084  df-tru 1539  df-ex 1780  df-nf 1784  df-sb 2069  df-clab 2799  df-cleq 2813  df-clel 2892  df-nfc 2962  df-ral 3142  df-rex 3143  df-v 3493  df-sbc 3769  df-un 3934  df-in 3936  df-ss 3945  df-sn 4561  df-pr 4563  df-iun 4914

This theorem is referenced by: (None)