Theorem iunlub 49442

Description: The indexed union is the the lowest upper bound if it exists. (Contributed by Zhi Wang, 1-Nov-2025.)

Hypotheses

Ref	Expression
iunlub.1	⊢ (𝜑 → 𝑋 ∈ 𝐴)
iunlub.2	⊢ ((𝜑 ∧ 𝑥 = 𝑋) → 𝐵 = 𝐶)
iunlub.3	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ⊆ 𝐶)

Assertion

Ref	Expression
iunlub	⊢ (𝜑 → ∪ 𝑥 ∈ 𝐴 𝐵 = 𝐶)

Distinct variable groups:   𝑥,𝐴   𝑥,𝐶   𝑥,𝑋   𝜑,𝑥

Allowed substitution hint:   𝐵(𝑥)

Proof of Theorem iunlub

Step	Hyp	Ref	Expression
1		iunlub.3	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ⊆ 𝐶)
2	1	iunssd 5008	. 2 ⊢ (𝜑 → ∪ 𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶)
3		iunlub.1	. . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝐴)
4		iunlub.2	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 = 𝑋) → 𝐵 = 𝐶)
5	4	sseq2d 3968	. . . 4 ⊢ ((𝜑 ∧ 𝑥 = 𝑋) → (𝐶 ⊆ 𝐵 ↔ 𝐶 ⊆ 𝐶))
6		ssidd 3959	. . . 4 ⊢ (𝜑 → 𝐶 ⊆ 𝐶)
7	3, 5, 6	rspcedvd 3583	. . 3 ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 𝐶 ⊆ 𝐵)
8		ssiun 5004	. . 3 ⊢ (∃𝑥 ∈ 𝐴 𝐶 ⊆ 𝐵 → 𝐶 ⊆ ∪ 𝑥 ∈ 𝐴 𝐵)
9	7, 8	syl 17	. 2 ⊢ (𝜑 → 𝐶 ⊆ ∪ 𝑥 ∈ 𝐴 𝐵)
10	2, 9	eqssd 3953	1 ⊢ (𝜑 → ∪ 𝑥 ∈ 𝐴 𝐵 = 𝐶)

Syntax hints:   → wi 4   ∧ wa 399   = wceq 1560   ∈ wcel 2142  ∃wrex 3086   ⊆ wss 3904  ∪ ciun 4949

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-8 2144  ax-9 2152  ax-11 2191  ax-ext 2734

This theorem depends on definitions:  df-bi 209  df-an 400  df-tru 1563  df-ex 1800  df-sb 2091  df-clab 2741  df-cleq 2754  df-clel 2837  df-ral 3077  df-rex 3087  df-v 3456  df-ss 3921  df-iun 4951

This theorem is referenced by: (None)