Theorem iunxunsn 32589

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

Hypothesis

Ref	Expression
iunxunsn.1	⊢ (𝑥 = 𝑋 → 𝐵 = 𝐶)

Assertion

Ref	Expression
iunxunsn	⊢ (𝑋 ∈ 𝑉 → ∪ 𝑥 ∈ (𝐴 ∪ {𝑋})𝐵 = (∪ 𝑥 ∈ 𝐴 𝐵 ∪ 𝐶))

Distinct variable groups:   𝑥,𝐶   𝑥,𝑋

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

Proof of Theorem iunxunsn

Step	Hyp	Ref	Expression
1		iunxun 5117	. 2 ⊢ ∪ 𝑥 ∈ (𝐴 ∪ {𝑋})𝐵 = (∪ 𝑥 ∈ 𝐴 𝐵 ∪ ∪ 𝑥 ∈ {𝑋}𝐵)
2		iunxunsn.1	. . . 4 ⊢ (𝑥 = 𝑋 → 𝐵 = 𝐶)
3	2	iunxsng 5113	. . 3 ⊢ (𝑋 ∈ 𝑉 → ∪ 𝑥 ∈ {𝑋}𝐵 = 𝐶)
4	3	uneq2d 4191	. 2 ⊢ (𝑋 ∈ 𝑉 → (∪ 𝑥 ∈ 𝐴 𝐵 ∪ ∪ 𝑥 ∈ {𝑋}𝐵) = (∪ 𝑥 ∈ 𝐴 𝐵 ∪ 𝐶))
5	1, 4	eqtrid 2792	1 ⊢ (𝑋 ∈ 𝑉 → ∪ 𝑥 ∈ (𝐴 ∪ {𝑋})𝐵 = (∪ 𝑥 ∈ 𝐴 𝐵 ∪ 𝐶))

Syntax hints:   → wi 4   = wceq 1537   ∈ wcel 2108   ∪ cun 3974  {csn 4648  ∪ ciun 5015

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 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2711

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-tru 1540  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-ral 3068  df-rex 3077  df-v 3490  df-un 3981  df-sn 4649  df-iun 5017

This theorem is referenced by: (None)