Theorem bj-restuni2 37099

Description: The union of an elementwise intersection on a family of sets by a subset is equal to that subset. See also restuni 23170 and restuni2 23175. (Contributed by BJ, 27-Apr-2021.)

Assertion

Ref	Expression
bj-restuni2	⊢ ((𝑋 ∈ 𝑉 ∧ 𝐴 ⊆ ∪ 𝑋) → ∪ (𝑋 ↾t 𝐴) = 𝐴)

Proof of Theorem bj-restuni2

Step	Hyp	Ref	Expression
1		uniexg 7760	. . . . 5 ⊢ (𝑋 ∈ 𝑉 → ∪ 𝑋 ∈ V)
2		ssexg 5323	. . . . 5 ⊢ ((𝐴 ⊆ ∪ 𝑋 ∧ ∪ 𝑋 ∈ V) → 𝐴 ∈ V)
3	1, 2	sylan2 593	. . . 4 ⊢ ((𝐴 ⊆ ∪ 𝑋 ∧ 𝑋 ∈ 𝑉) → 𝐴 ∈ V)
4	3	ancoms 458	. . 3 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝐴 ⊆ ∪ 𝑋) → 𝐴 ∈ V)
5		bj-restuni 37098	. . 3 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝐴 ∈ V) → ∪ (𝑋 ↾t 𝐴) = (∪ 𝑋 ∩ 𝐴))
6	4, 5	syldan 591	. 2 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝐴 ⊆ ∪ 𝑋) → ∪ (𝑋 ↾t 𝐴) = (∪ 𝑋 ∩ 𝐴))
7		inss2 4238	. . . . 5 ⊢ (∪ 𝑋 ∩ 𝐴) ⊆ 𝐴
8	7	a1i 11	. . . 4 ⊢ (𝐴 ⊆ ∪ 𝑋 → (∪ 𝑋 ∩ 𝐴) ⊆ 𝐴)
9		id 22	. . . . 5 ⊢ (𝐴 ⊆ ∪ 𝑋 → 𝐴 ⊆ ∪ 𝑋)
10		ssidd 4007	. . . . 5 ⊢ (𝐴 ⊆ ∪ 𝑋 → 𝐴 ⊆ 𝐴)
11	9, 10	ssind 4241	. . . 4 ⊢ (𝐴 ⊆ ∪ 𝑋 → 𝐴 ⊆ (∪ 𝑋 ∩ 𝐴))
12	8, 11	eqssd 4001	. . 3 ⊢ (𝐴 ⊆ ∪ 𝑋 → (∪ 𝑋 ∩ 𝐴) = 𝐴)
13	12	adantl 481	. 2 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝐴 ⊆ ∪ 𝑋) → (∪ 𝑋 ∩ 𝐴) = 𝐴)
14	6, 13	eqtrd 2777	1 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝐴 ⊆ ∪ 𝑋) → ∪ (𝑋 ↾t 𝐴) = 𝐴)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2108  Vcvv 3480   ∩ cin 3950   ⊆ wss 3951  ∪ cuni 4907  (class class class)co 7431   ↾_t crest 17465

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 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pr 5432  ax-un 7755

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-rest 17467

This theorem is referenced by: (None)