Theorem iuneqconst2 48804

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 4002	. . . . 5 ⊢ (𝐵 = 𝐶 → 𝐵 ⊆ 𝐶)
2	1	ralimi 3066	. . . 4 ⊢ (∀𝑥 ∈ 𝐴 𝐵 = 𝐶 → ∀𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶)
3	2	adantl 481	. . 3 ⊢ ((𝐴 ≠ ∅ ∧ ∀𝑥 ∈ 𝐴 𝐵 = 𝐶) → ∀𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶)
4		iunss 5004	. . 3 ⊢ (∪ 𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶 ↔ ∀𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶)
5	3, 4	sylibr 234	. 2 ⊢ ((𝐴 ≠ ∅ ∧ ∀𝑥 ∈ 𝐴 𝐵 = 𝐶) → ∪ 𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶)
6		r19.2z 4454	. . 3 ⊢ ((𝐴 ≠ ∅ ∧ ∀𝑥 ∈ 𝐴 𝐵 = 𝐶) → ∃𝑥 ∈ 𝐴 𝐵 = 𝐶)
7		eqimss2 4003	. . . 4 ⊢ (𝐵 = 𝐶 → 𝐶 ⊆ 𝐵)
8	7	reximi 3067	. . 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 18	. 2 ⊢ ((𝐴 ≠ ∅ ∧ ∀𝑥 ∈ 𝐴 𝐵 = 𝐶) → 𝐶 ⊆ ∪ 𝑥 ∈ 𝐴 𝐵)
11	5, 10	eqssd 3961	1 ⊢ ((𝐴 ≠ ∅ ∧ ∀𝑥 ∈ 𝐴 𝐵 = 𝐶) → ∪ 𝑥 ∈ 𝐴 𝐵 = 𝐶)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ≠ wne 2925  ∀wral 3044  ∃wrex 3053   ⊆ wss 3911  ∅c0 4292  ∪ ciun 4951

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-v 3446  df-dif 3914  df-ss 3928  df-nul 4293  df-iun 4953

This theorem is referenced by:  imasubc  49133