Theorem bj-restuni2 35269

Description: The union of an elementwise intersection on a family of sets by a subset is equal to that subset. See also restuni 22313 and restuni2 22318. (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 7593	. . . . 5 ⊢ (𝑋 ∈ 𝑉 → ∪ 𝑋 ∈ V)
2		ssexg 5247	. . . . 5 ⊢ ((𝐴 ⊆ ∪ 𝑋 ∧ ∪ 𝑋 ∈ V) → 𝐴 ∈ V)
3	1, 2	sylan2 593	. . . 4 ⊢ ((𝐴 ⊆ ∪ 𝑋 ∧ 𝑋 ∈ 𝑉) → 𝐴 ∈ V)
4	3	ancoms 459	. . 3 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝐴 ⊆ ∪ 𝑋) → 𝐴 ∈ V)
5		bj-restuni 35268	. . 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 4163	. . . . 5 ⊢ (∪ 𝑋 ∩ 𝐴) ⊆ 𝐴
8	7	a1i 11	. . . 4 ⊢ (𝐴 ⊆ ∪ 𝑋 → (∪ 𝑋 ∩ 𝐴) ⊆ 𝐴)
9		id 22	. . . . 5 ⊢ (𝐴 ⊆ ∪ 𝑋 → 𝐴 ⊆ ∪ 𝑋)
10		ssidd 3944	. . . . 5 ⊢ (𝐴 ⊆ ∪ 𝑋 → 𝐴 ⊆ 𝐴)
11	9, 10	ssind 4166	. . . 4 ⊢ (𝐴 ⊆ ∪ 𝑋 → 𝐴 ⊆ (∪ 𝑋 ∩ 𝐴))
12	8, 11	eqssd 3938	. . 3 ⊢ (𝐴 ⊆ ∪ 𝑋 → (∪ 𝑋 ∩ 𝐴) = 𝐴)
13	12	adantl 482	. 2 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝐴 ⊆ ∪ 𝑋) → (∪ 𝑋 ∩ 𝐴) = 𝐴)
14	6, 13	eqtrd 2778	1 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝐴 ⊆ ∪ 𝑋) → ∪ (𝑋 ↾t 𝐴) = 𝐴)

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1539   ∈ wcel 2106  Vcvv 3432   ∩ cin 3886   ⊆ wss 3887  ∪ cuni 4839  (class class class)co 7275   ↾_t crest 17131

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pr 5352  ax-un 7588

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-ov 7278  df-oprab 7279  df-mpo 7280  df-rest 17133

This theorem is referenced by: (None)