Theorem ralssiun 36592

Description: The index set of an indexed union is a subset of the union when each 𝐵 contains its index. (Contributed by ML, 16-Dec-2020.)

Assertion

Ref	Expression
ralssiun	⊢ (∀𝑥 ∈ 𝐴 𝑥 ∈ 𝐵 → 𝐴 ⊆ ∪ 𝑥 ∈ 𝐴 𝐵)

Distinct variable group:   𝑥,𝐴

Allowed substitution hint:   𝐵(𝑥)

Proof of Theorem ralssiun

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		nfra1 3280	. 2 ⊢ Ⅎ𝑥∀𝑥 ∈ 𝐴 𝑥 ∈ 𝐵
2		nfcv 2902	. 2 ⊢ Ⅎ𝑥𝐴
3		nfiu1 5031	. 2 ⊢ Ⅎ𝑥∪ 𝑥 ∈ 𝐴 𝐵
4		simpr 484	. . . . . . . . . . 11 ⊢ (((𝑥 = 𝑦 ∧ ∀𝑥 ∈ 𝐴 𝑥 ∈ 𝐵) ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ 𝐴)
5		rsp 3243	. . . . . . . . . . . . . 14 ⊢ (∀𝑥 ∈ 𝐴 𝑥 ∈ 𝐵 → (𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐵))
6	5	adantl 481	. . . . . . . . . . . . 13 ⊢ ((𝑥 = 𝑦 ∧ ∀𝑥 ∈ 𝐴 𝑥 ∈ 𝐵) → (𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐵))
7		eleq1 2820	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝐵 ↔ 𝑦 ∈ 𝐵))
8	7	imbi2d 340	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑦 → ((𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐵) ↔ (𝑥 ∈ 𝐴 → 𝑦 ∈ 𝐵)))
9	8	adantr 480	. . . . . . . . . . . . 13 ⊢ ((𝑥 = 𝑦 ∧ ∀𝑥 ∈ 𝐴 𝑥 ∈ 𝐵) → ((𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐵) ↔ (𝑥 ∈ 𝐴 → 𝑦 ∈ 𝐵)))
10	6, 9	mpbid 231	. . . . . . . . . . . 12 ⊢ ((𝑥 = 𝑦 ∧ ∀𝑥 ∈ 𝐴 𝑥 ∈ 𝐵) → (𝑥 ∈ 𝐴 → 𝑦 ∈ 𝐵))
11	10	imp 406	. . . . . . . . . . 11 ⊢ (((𝑥 = 𝑦 ∧ ∀𝑥 ∈ 𝐴 𝑥 ∈ 𝐵) ∧ 𝑥 ∈ 𝐴) → 𝑦 ∈ 𝐵)
12		rspe 3245	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐵)
13	4, 11, 12	syl2anc 583	. . . . . . . . . 10 ⊢ (((𝑥 = 𝑦 ∧ ∀𝑥 ∈ 𝐴 𝑥 ∈ 𝐵) ∧ 𝑥 ∈ 𝐴) → ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐵)
14		abid 2712	. . . . . . . . . 10 ⊢ (𝑦 ∈ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐵} ↔ ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐵)
15	13, 14	sylibr 233	. . . . . . . . 9 ⊢ (((𝑥 = 𝑦 ∧ ∀𝑥 ∈ 𝐴 𝑥 ∈ 𝐵) ∧ 𝑥 ∈ 𝐴) → 𝑦 ∈ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐵})
16		eleq1 2820	. . . . . . . . . 10 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐵} ↔ 𝑦 ∈ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐵}))
17	16	ad2antrr 723	. . . . . . . . 9 ⊢ (((𝑥 = 𝑦 ∧ ∀𝑥 ∈ 𝐴 𝑥 ∈ 𝐵) ∧ 𝑥 ∈ 𝐴) → (𝑥 ∈ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐵} ↔ 𝑦 ∈ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐵}))
18	15, 17	mpbird 257	. . . . . . . 8 ⊢ (((𝑥 = 𝑦 ∧ ∀𝑥 ∈ 𝐴 𝑥 ∈ 𝐵) ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐵})
19		df-iun 4999	. . . . . . . 8 ⊢ ∪ 𝑥 ∈ 𝐴 𝐵 = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐵}
20	18, 19	eleqtrrdi 2843	. . . . . . 7 ⊢ (((𝑥 = 𝑦 ∧ ∀𝑥 ∈ 𝐴 𝑥 ∈ 𝐵) ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ ∪ 𝑥 ∈ 𝐴 𝐵)
21	20	expl 457	. . . . . 6 ⊢ (𝑥 = 𝑦 → ((∀𝑥 ∈ 𝐴 𝑥 ∈ 𝐵 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ ∪ 𝑥 ∈ 𝐴 𝐵))
22	21	equcoms 2022	. . . . 5 ⊢ (𝑦 = 𝑥 → ((∀𝑥 ∈ 𝐴 𝑥 ∈ 𝐵 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ ∪ 𝑥 ∈ 𝐴 𝐵))
23	22	vtocleg 3541	. . . 4 ⊢ (𝑥 ∈ 𝐴 → ((∀𝑥 ∈ 𝐴 𝑥 ∈ 𝐵 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ ∪ 𝑥 ∈ 𝐴 𝐵))
24	23	anabsi7 668	. . 3 ⊢ ((∀𝑥 ∈ 𝐴 𝑥 ∈ 𝐵 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ ∪ 𝑥 ∈ 𝐴 𝐵)
25	24	ex 412	. 2 ⊢ (∀𝑥 ∈ 𝐴 𝑥 ∈ 𝐵 → (𝑥 ∈ 𝐴 → 𝑥 ∈ ∪ 𝑥 ∈ 𝐴 𝐵))
26	1, 2, 3, 25	ssrd 3987	1 ⊢ (∀𝑥 ∈ 𝐴 𝑥 ∈ 𝐵 → 𝐴 ⊆ ∪ 𝑥 ∈ 𝐴 𝐵)

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   = wceq 1540   ∈ wcel 2105  {cab 2708  ∀wral 3060  ∃wrex 3069   ⊆ wss 3948  ∪ ciun 4997

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 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-tru 1543  df-ex 1781  df-nf 1785  df-sb 2067  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ral 3061  df-rex 3070  df-v 3475  df-in 3955  df-ss 3965  df-iun 4999

This theorem is referenced by:  pibt2  36602