Theorem iunxunsn 32546

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 5040	. 2 ⊢ ∪ 𝑥 ∈ (𝐴 ∪ {𝑋})𝐵 = (∪ 𝑥 ∈ 𝐴 𝐵 ∪ ∪ 𝑥 ∈ {𝑋}𝐵)
2		iunxunsn.1	. . . 4 ⊢ (𝑥 = 𝑋 → 𝐵 = 𝐶)
3	2	iunxsng 5036	. . 3 ⊢ (𝑋 ∈ 𝑉 → ∪ 𝑥 ∈ {𝑋}𝐵 = 𝐶)
4	3	uneq2d 4115	. 2 ⊢ (𝑋 ∈ 𝑉 → (∪ 𝑥 ∈ 𝐴 𝐵 ∪ ∪ 𝑥 ∈ {𝑋}𝐵) = (∪ 𝑥 ∈ 𝐴 𝐵 ∪ 𝐶))
5	1, 4	eqtrid 2778	1 ⊢ (𝑋 ∈ 𝑉 → ∪ 𝑥 ∈ (𝐴 ∪ {𝑋})𝐵 = (∪ 𝑥 ∈ 𝐴 𝐵 ∪ 𝐶))

Syntax hints:   → wi 4   = wceq 1541   ∈ wcel 2111   ∪ cun 3895  {csn 4573  ∪ ciun 4939

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-ext 2703

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1544  df-ex 1781  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-ral 3048  df-rex 3057  df-v 3438  df-un 3902  df-sn 4574  df-iun 4941

This theorem is referenced by: (None)