Theorem bj-restuni2 37215

Description: The union of an elementwise intersection on a family of sets by a subset is equal to that subset. See also restuni 23097 and restuni2 23102. (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 7682	. . . . 5 ⊢ (𝑋 ∈ 𝑉 → ∪ 𝑋 ∈ V)
2		ssexg 5265	. . . . 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 37214	. . 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 4187	. . . . 5 ⊢ (∪ 𝑋 ∩ 𝐴) ⊆ 𝐴
8	7	a1i 11	. . . 4 ⊢ (𝐴 ⊆ ∪ 𝑋 → (∪ 𝑋 ∩ 𝐴) ⊆ 𝐴)
9		id 22	. . . . 5 ⊢ (𝐴 ⊆ ∪ 𝑋 → 𝐴 ⊆ ∪ 𝑋)
10		ssidd 3954	. . . . 5 ⊢ (𝐴 ⊆ ∪ 𝑋 → 𝐴 ⊆ 𝐴)
11	9, 10	ssind 4190	. . . 4 ⊢ (𝐴 ⊆ ∪ 𝑋 → 𝐴 ⊆ (∪ 𝑋 ∩ 𝐴))
12	8, 11	eqssd 3948	. . 3 ⊢ (𝐴 ⊆ ∪ 𝑋 → (∪ 𝑋 ∩ 𝐴) = 𝐴)
13	12	adantl 481	. 2 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝐴 ⊆ ∪ 𝑋) → (∪ 𝑋 ∩ 𝐴) = 𝐴)
14	6, 13	eqtrd 2768	1 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝐴 ⊆ ∪ 𝑋) → ∪ (𝑋 ↾t 𝐴) = 𝐴)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1541   ∈ wcel 2113  Vcvv 3437   ∩ cin 3897   ⊆ wss 3898  ∪ cuni 4860  (class class class)co 7355   ↾_t crest 17331

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 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2705  ax-rep 5221  ax-sep 5238  ax-nul 5248  ax-pr 5374  ax-un 7677

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2725  df-clel 2808  df-nfc 2882  df-ne 2930  df-ral 3049  df-rex 3058  df-reu 3348  df-rab 3397  df-v 3439  df-sbc 3738  df-csb 3847  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4283  df-if 4477  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4861  df-iun 4945  df-br 5096  df-opab 5158  df-mpt 5177  df-id 5516  df-xp 5627  df-rel 5628  df-cnv 5629  df-co 5630  df-dm 5631  df-rn 5632  df-res 5633  df-ima 5634  df-iota 6445  df-fun 6491  df-fn 6492  df-f 6493  df-f1 6494  df-fo 6495  df-f1o 6496  df-fv 6497  df-ov 7358  df-oprab 7359  df-mpo 7360  df-rest 17333

This theorem is referenced by: (None)