Theorem bj-rest0 34948

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 4292	. . . . 5 ⊢ (𝐴 ∩ ∅) = ∅
2		incom 4101	. . . . 5 ⊢ (𝐴 ∩ ∅) = (∅ ∩ 𝐴)
3	1, 2	eqtr3i 2761	. . . 4 ⊢ ∅ = (∅ ∩ 𝐴)
4		0ex 5185	. . . . 5 ⊢ ∅ ∈ V
5		eleq1 2818	. . . . . 6 ⊢ (𝑥 = ∅ → (𝑥 ∈ 𝑋 ↔ ∅ ∈ 𝑋))
6		ineq1 4106	. . . . . . 7 ⊢ (𝑥 = ∅ → (𝑥 ∩ 𝐴) = (∅ ∩ 𝐴))
7	6	eqeq2d 2747	. . . . . 6 ⊢ (𝑥 = ∅ → (∅ = (𝑥 ∩ 𝐴) ↔ ∅ = (∅ ∩ 𝐴)))
8	5, 7	anbi12d 634	. . . . 5 ⊢ (𝑥 = ∅ → ((𝑥 ∈ 𝑋 ∧ ∅ = (𝑥 ∩ 𝐴)) ↔ (∅ ∈ 𝑋 ∧ ∅ = (∅ ∩ 𝐴))))
9	4, 8	spcev 3511	. . . 4 ⊢ ((∅ ∈ 𝑋 ∧ ∅ = (∅ ∩ 𝐴)) → ∃𝑥(𝑥 ∈ 𝑋 ∧ ∅ = (𝑥 ∩ 𝐴)))
10	3, 9	mpan2 691	. . 3 ⊢ (∅ ∈ 𝑋 → ∃𝑥(𝑥 ∈ 𝑋 ∧ ∅ = (𝑥 ∩ 𝐴)))
11		df-rex 3057	. . 3 ⊢ (∃𝑥 ∈ 𝑋 ∅ = (𝑥 ∩ 𝐴) ↔ ∃𝑥(𝑥 ∈ 𝑋 ∧ ∅ = (𝑥 ∩ 𝐴)))
12	10, 11	sylibr 237	. 2 ⊢ (∅ ∈ 𝑋 → ∃𝑥 ∈ 𝑋 ∅ = (𝑥 ∩ 𝐴))
13		elrest 16886	. 2 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊) → (∅ ∈ (𝑋 ↾t 𝐴) ↔ ∃𝑥 ∈ 𝑋 ∅ = (𝑥 ∩ 𝐴)))
14	12, 13	syl5ibr 249	1 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊) → (∅ ∈ 𝑋 → ∅ ∈ (𝑋 ↾t 𝐴)))

Syntax hints:   → wi 4   ∧ wa 399   = wceq 1543  ∃wex 1787   ∈ wcel 2112  ∃wrex 3052   ∩ cin 3852  ∅c0 4223  (class class class)co 7191   ↾_t crest 16879

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2018  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2160  ax-12 2177  ax-ext 2708  ax-rep 5164  ax-sep 5177  ax-nul 5184  ax-pr 5307  ax-un 7501

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2073  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2728  df-clel 2809  df-nfc 2879  df-ne 2933  df-ral 3056  df-rex 3057  df-reu 3058  df-rab 3060  df-v 3400  df-sbc 3684  df-csb 3799  df-dif 3856  df-un 3858  df-in 3860  df-ss 3870  df-nul 4224  df-if 4426  df-sn 4528  df-pr 4530  df-op 4534  df-uni 4806  df-iun 4892  df-br 5040  df-opab 5102  df-mpt 5121  df-id 5440  df-xp 5542  df-rel 5543  df-cnv 5544  df-co 5545  df-dm 5546  df-rn 5547  df-res 5548  df-ima 5549  df-iota 6316  df-fun 6360  df-fn 6361  df-f 6362  df-f1 6363  df-fo 6364  df-f1o 6365  df-fv 6366  df-ov 7194  df-oprab 7195  df-mpo 7196  df-rest 16881

This theorem is referenced by: (None)