Theorem iunxunsn 32233

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 5087	. 2 ⊢ ∪ 𝑥 ∈ (𝐴 ∪ {𝑋})𝐵 = (∪ 𝑥 ∈ 𝐴 𝐵 ∪ ∪ 𝑥 ∈ {𝑋}𝐵)
2		iunxunsn.1	. . . 4 ⊢ (𝑥 = 𝑋 → 𝐵 = 𝐶)
3	2	iunxsng 5083	. . 3 ⊢ (𝑋 ∈ 𝑉 → ∪ 𝑥 ∈ {𝑋}𝐵 = 𝐶)
4	3	uneq2d 4155	. 2 ⊢ (𝑋 ∈ 𝑉 → (∪ 𝑥 ∈ 𝐴 𝐵 ∪ ∪ 𝑥 ∈ {𝑋}𝐵) = (∪ 𝑥 ∈ 𝐴 𝐵 ∪ 𝐶))
5	1, 4	eqtrid 2776	1 ⊢ (𝑋 ∈ 𝑉 → ∪ 𝑥 ∈ (𝐴 ∪ {𝑋})𝐵 = (∪ 𝑥 ∈ 𝐴 𝐵 ∪ 𝐶))

Syntax hints:   → wi 4   = wceq 1533   ∈ wcel 2098   ∪ cun 3938  {csn 4620  ∪ ciun 4987

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2695

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-tru 1536  df-ex 1774  df-sb 2060  df-clab 2702  df-cleq 2716  df-clel 2802  df-ral 3054  df-rex 3063  df-v 3468  df-un 3945  df-sn 4621  df-iun 4989

This theorem is referenced by: (None)