Theorem bj-rest0 36278

Description: An elementwise intersection on a family containing the empty set contains the empty set. (Contributed by BJ, 27-Apr-2021.)

Assertion

Ref	Expression
bj-rest0	⊢ ((𝑋 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊) → (∅ ∈ 𝑋 → ∅ ∈ (𝑋 ↾t 𝐴)))

Proof of Theorem bj-rest0

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		in0 4391	. . . . 5 ⊢ (𝐴 ∩ ∅) = ∅
2		incom 4201	. . . . 5 ⊢ (𝐴 ∩ ∅) = (∅ ∩ 𝐴)
3	1, 2	eqtr3i 2761	. . . 4 ⊢ ∅ = (∅ ∩ 𝐴)
4		0ex 5307	. . . . 5 ⊢ ∅ ∈ V
5		eleq1 2820	. . . . . 6 ⊢ (𝑥 = ∅ → (𝑥 ∈ 𝑋 ↔ ∅ ∈ 𝑋))
6		ineq1 4205	. . . . . . 7 ⊢ (𝑥 = ∅ → (𝑥 ∩ 𝐴) = (∅ ∩ 𝐴))
7	6	eqeq2d 2742	. . . . . 6 ⊢ (𝑥 = ∅ → (∅ = (𝑥 ∩ 𝐴) ↔ ∅ = (∅ ∩ 𝐴)))
8	5, 7	anbi12d 630	. . . . 5 ⊢ (𝑥 = ∅ → ((𝑥 ∈ 𝑋 ∧ ∅ = (𝑥 ∩ 𝐴)) ↔ (∅ ∈ 𝑋 ∧ ∅ = (∅ ∩ 𝐴))))
9	4, 8	spcev 3596	. . . 4 ⊢ ((∅ ∈ 𝑋 ∧ ∅ = (∅ ∩ 𝐴)) → ∃𝑥(𝑥 ∈ 𝑋 ∧ ∅ = (𝑥 ∩ 𝐴)))
10	3, 9	mpan2 688	. . 3 ⊢ (∅ ∈ 𝑋 → ∃𝑥(𝑥 ∈ 𝑋 ∧ ∅ = (𝑥 ∩ 𝐴)))
11		df-rex 3070	. . 3 ⊢ (∃𝑥 ∈ 𝑋 ∅ = (𝑥 ∩ 𝐴) ↔ ∃𝑥(𝑥 ∈ 𝑋 ∧ ∅ = (𝑥 ∩ 𝐴)))
12	10, 11	sylibr 233	. 2 ⊢ (∅ ∈ 𝑋 → ∃𝑥 ∈ 𝑋 ∅ = (𝑥 ∩ 𝐴))
13		elrest 17378	. 2 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊) → (∅ ∈ (𝑋 ↾t 𝐴) ↔ ∃𝑥 ∈ 𝑋 ∅ = (𝑥 ∩ 𝐴)))
14	12, 13	imbitrrid 245	1 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊) → (∅ ∈ 𝑋 → ∅ ∈ (𝑋 ↾t 𝐴)))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540  ∃wex 1780   ∈ wcel 2105  ∃wrex 3069   ∩ cin 3947  ∅c0 4322  (class class class)co 7412   ↾_t crest 17371

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  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pr 5427  ax-un 7728

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-reu 3376  df-rab 3432  df-v 3475  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-ov 7415  df-oprab 7416  df-mpo 7417  df-rest 17373

This theorem is referenced by: (None)