Theorem iuneqconst2 49754

Description: Indexed union of identical classes. (Contributed by Zhi Wang, 6-Nov-2025.)

Assertion

Ref	Expression
iuneqconst2	⊢ ((𝐴 ≠ ∅ ∧ ∀𝑥 ∈ 𝐴 𝐵 = 𝐶) → ∪ 𝑥 ∈ 𝐴 𝐵 = 𝐶)

Distinct variable groups:   𝑥,𝐴   𝑥,𝐶

Allowed substitution hint:   𝐵(𝑥)

Proof of Theorem iuneqconst2

Step	Hyp	Ref	Expression
1		eqimss 3989	. . . . 5 ⊢ (𝐵 = 𝐶 → 𝐵 ⊆ 𝐶)
2	1	ralimi 3099	. . . 4 ⊢ (∀𝑥 ∈ 𝐴 𝐵 = 𝐶 → ∀𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶)
3	2	adantl 487	. . 3 ⊢ ((𝐴 ≠ ∅ ∧ ∀𝑥 ∈ 𝐴 𝐵 = 𝐶) → ∀𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶)
4		iunss 5003	. . 3 ⊢ (∪ 𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶 ↔ ∀𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶)
5	3, 4	sylibr 237	. 2 ⊢ ((𝐴 ≠ ∅ ∧ ∀𝑥 ∈ 𝐴 𝐵 = 𝐶) → ∪ 𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶)
6		r19.2z 4455	. . 3 ⊢ ((𝐴 ≠ ∅ ∧ ∀𝑥 ∈ 𝐴 𝐵 = 𝐶) → ∃𝑥 ∈ 𝐴 𝐵 = 𝐶)
7		eqimss2 3990	. . . 4 ⊢ (𝐵 = 𝐶 → 𝐶 ⊆ 𝐵)
8	7	reximi 3100	. . 3 ⊢ (∃𝑥 ∈ 𝐴 𝐵 = 𝐶 → ∃𝑥 ∈ 𝐴 𝐶 ⊆ 𝐵)
9		ssiun 5005	. . 3 ⊢ (∃𝑥 ∈ 𝐴 𝐶 ⊆ 𝐵 → 𝐶 ⊆ ∪ 𝑥 ∈ 𝐴 𝐵)
10	6, 8, 9	3syl 19	. 2 ⊢ ((𝐴 ≠ ∅ ∧ ∀𝑥 ∈ 𝐴 𝐵 = 𝐶) → 𝐶 ⊆ ∪ 𝑥 ∈ 𝐴 𝐵)
11	5, 10	eqssd 3948	1 ⊢ ((𝐴 ≠ ∅ ∧ ∀𝑥 ∈ 𝐴 𝐵 = 𝐶) → ∪ 𝑥 ∈ 𝐴 𝐵 = 𝐶)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ≠ wne 2955  ∀wral 3076  ∃wrex 3086   ⊆ wss 3899  ∅c0 4279  ∪ ciun 4951

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 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-11 2194  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2739  df-cleq 2752  df-clel 2835  df-ne 2956  df-ral 3077  df-rex 3087  df-v 3452  df-dif 3902  df-ss 3916  df-nul 4280  df-iun 4953

This theorem is used by:  imasubc  50080