Theorem bj-restuni2 37086

Description: The union of an elementwise intersection on a family of sets by a subset is equal to that subset. See also restuni 23049 and restuni2 23054. (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 7716	. . . . 5 ⊢ (𝑋 ∈ 𝑉 → ∪ 𝑋 ∈ V)
2		ssexg 5278	. . . . 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 37085	. . 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 4201	. . . . 5 ⊢ (∪ 𝑋 ∩ 𝐴) ⊆ 𝐴
8	7	a1i 11	. . . 4 ⊢ (𝐴 ⊆ ∪ 𝑋 → (∪ 𝑋 ∩ 𝐴) ⊆ 𝐴)
9		id 22	. . . . 5 ⊢ (𝐴 ⊆ ∪ 𝑋 → 𝐴 ⊆ ∪ 𝑋)
10		ssidd 3970	. . . . 5 ⊢ (𝐴 ⊆ ∪ 𝑋 → 𝐴 ⊆ 𝐴)
11	9, 10	ssind 4204	. . . 4 ⊢ (𝐴 ⊆ ∪ 𝑋 → 𝐴 ⊆ (∪ 𝑋 ∩ 𝐴))
12	8, 11	eqssd 3964	. . 3 ⊢ (𝐴 ⊆ ∪ 𝑋 → (∪ 𝑋 ∩ 𝐴) = 𝐴)
13	12	adantl 481	. 2 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝐴 ⊆ ∪ 𝑋) → (∪ 𝑋 ∩ 𝐴) = 𝐴)
14	6, 13	eqtrd 2764	1 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝐴 ⊆ ∪ 𝑋) → ∪ (𝑋 ↾t 𝐴) = 𝐴)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109  Vcvv 3447   ∩ cin 3913   ⊆ wss 3914  ∪ cuni 4871  (class class class)co 7387   ↾_t crest 17383

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  ax-rep 5234  ax-sep 5251  ax-nul 5261  ax-pr 5387  ax-un 7711

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-reu 3355  df-rab 3406  df-v 3449  df-sbc 3754  df-csb 3863  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4297  df-if 4489  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-iun 4957  df-br 5108  df-opab 5170  df-mpt 5189  df-id 5533  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-iota 6464  df-fun 6513  df-fn 6514  df-f 6515  df-f1 6516  df-fo 6517  df-f1o 6518  df-fv 6519  df-ov 7390  df-oprab 7391  df-mpo 7392  df-rest 17385

This theorem is referenced by: (None)