Theorem iunxunsn 33229

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 5057	. 2 ⊢ ∪ 𝑥 ∈ (𝐴 ∪ {𝑋})𝐵 = (∪ 𝑥 ∈ 𝐴 𝐵 ∪ ∪ 𝑥 ∈ {𝑋}𝐵)
2		iunxunsn.1	. . . 4 ⊢ (𝑥 = 𝑋 → 𝐵 = 𝐶)
3	2	iunxsng 5053	. . 3 ⊢ (𝑋 ∈ 𝑉 → ∪ 𝑥 ∈ {𝑋}𝐵 = 𝐶)
4	3	uneq2d 4170	. 2 ⊢ (𝑋 ∈ 𝑉 → (∪ 𝑥 ∈ 𝐴 𝐵 ∪ ∪ 𝑥 ∈ {𝑋}𝐵) = (∪ 𝑥 ∈ 𝐴 𝐵 ∪ 𝐶))
5	1, 4	eqtrid 2810	1 ⊢ (𝑋 ∈ 𝑉 → ∪ 𝑥 ∈ (𝐴 ∪ {𝑋})𝐵 = (∪ 𝑥 ∈ 𝐴 𝐵 ∪ 𝐶))

This proof depends on syntax axioms:   → wi 4   = wceq 1570   ∈ wcel 2147   ∪ cun 4152  {csn 4588  ∪ ciun 4954

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1573  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-ral 3081  df-rex 3091  df-v 3456  df-un 4154  df-sn 4589  df-iun 4956

This theorem is used by: (None)